xref: /petsc/src/mat/impls/aij/seq/matmatmult.c (revision c524e44ceafee4aa1f13666ee553c16f4e41b8f4)
1 
2 /*
3   Defines matrix-matrix product routines for pairs of SeqAIJ matrices
4           C = A * B
5 */
6 
7 #include <../src/mat/impls/aij/seq/aij.h> /*I "petscmat.h" I*/
8 #include <../src/mat/utils/freespace.h>
9 #include <petscbt.h>
10 #include <petsc/private/isimpl.h>
11 #include <../src/mat/impls/dense/seq/dense.h>
12 
13 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
14 {
15   PetscErrorCode ierr;
16 
17   PetscFunctionBegin;
18   if (C->ops->matmultnumeric) {
19     if (C->ops->matmultnumeric == MatMatMultNumeric_SeqAIJ_SeqAIJ) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Recursive call");
20     ierr = (*C->ops->matmultnumeric)(A,B,C);CHKERRQ(ierr);
21   } else {
22     ierr = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted(A,B,C);CHKERRQ(ierr);
23   }
24   PetscFunctionReturn(0);
25 }
26 
27 /* Modified from MatCreateSeqAIJWithArrays() */
28 PETSC_INTERN PetscErrorCode MatSetSeqAIJWithArrays_private(MPI_Comm comm,PetscInt m,PetscInt n,PetscInt i[],PetscInt j[],PetscScalar a[],MatType mtype,Mat mat)
29 {
30   PetscErrorCode ierr;
31   PetscInt       ii;
32   Mat_SeqAIJ     *aij;
33   PetscBool      isseqaij;
34 
35   PetscFunctionBegin;
36   if (m > 0 && i[0]) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"i (row indices) must start with 0");
37   ierr = MatSetSizes(mat,m,n,m,n);CHKERRQ(ierr);
38 
39   if (!mtype) {
40     ierr = PetscObjectBaseTypeCompare((PetscObject)mat,MATSEQAIJ,&isseqaij);CHKERRQ(ierr);
41     if (!isseqaij) { ierr = MatSetType(mat,MATSEQAIJ);CHKERRQ(ierr); }
42   } else {
43     ierr = MatSetType(mat,mtype);CHKERRQ(ierr);
44   }
45   ierr = MatSeqAIJSetPreallocation_SeqAIJ(mat,MAT_SKIP_ALLOCATION,NULL);CHKERRQ(ierr);
46   aij  = (Mat_SeqAIJ*)(mat)->data;
47   ierr = PetscMalloc1(m,&aij->imax);CHKERRQ(ierr);
48   ierr = PetscMalloc1(m,&aij->ilen);CHKERRQ(ierr);
49 
50   aij->i            = i;
51   aij->j            = j;
52   aij->a            = a;
53   aij->singlemalloc = PETSC_FALSE;
54   aij->nonew        = -1; /*this indicates that inserting a new value in the matrix that generates a new nonzero is an error*/
55   aij->free_a       = PETSC_FALSE;
56   aij->free_ij      = PETSC_FALSE;
57 
58   for (ii=0; ii<m; ii++) {
59     aij->ilen[ii] = aij->imax[ii] = i[ii+1] - i[ii];
60   }
61 
62   PetscFunctionReturn(0);
63 }
64 
65 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat C)
66 {
67   PetscErrorCode      ierr;
68   Mat_Product         *product = C->product;
69   MatProductAlgorithm alg;
70   PetscBool           flg;
71 
72   PetscFunctionBegin;
73   if (product) {
74     alg = product->alg;
75   } else {
76     alg = "sorted";
77   }
78   /* sorted */
79   ierr = PetscStrcmp(alg,"sorted",&flg);CHKERRQ(ierr);
80   if (flg) {
81     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Sorted(A,B,fill,C);CHKERRQ(ierr);
82     PetscFunctionReturn(0);
83   }
84 
85   /* scalable */
86   ierr = PetscStrcmp(alg,"scalable",&flg);CHKERRQ(ierr);
87   if (flg) {
88     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable(A,B,fill,C);CHKERRQ(ierr);
89     PetscFunctionReturn(0);
90   }
91 
92   /* scalable_fast */
93   ierr = PetscStrcmp(alg,"scalable_fast",&flg);CHKERRQ(ierr);
94   if (flg) {
95     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable_fast(A,B,fill,C);CHKERRQ(ierr);
96     PetscFunctionReturn(0);
97   }
98 
99   /* heap */
100   ierr = PetscStrcmp(alg,"heap",&flg);CHKERRQ(ierr);
101   if (flg) {
102     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Heap(A,B,fill,C);CHKERRQ(ierr);
103     PetscFunctionReturn(0);
104   }
105 
106   /* btheap */
107   ierr = PetscStrcmp(alg,"btheap",&flg);CHKERRQ(ierr);
108   if (flg) {
109     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_BTHeap(A,B,fill,C);CHKERRQ(ierr);
110     PetscFunctionReturn(0);
111   }
112 
113   /* llcondensed */
114   ierr = PetscStrcmp(alg,"llcondensed",&flg);CHKERRQ(ierr);
115   if (flg) {
116     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(A,B,fill,C);CHKERRQ(ierr);
117     PetscFunctionReturn(0);
118   }
119 
120   /* rowmerge */
121   ierr = PetscStrcmp(alg,"rowmerge",&flg);CHKERRQ(ierr);
122   if (flg) {
123     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_RowMerge(A,B,fill,C);CHKERRQ(ierr);
124     PetscFunctionReturn(0);
125   }
126 
127 #if defined(PETSC_HAVE_HYPRE)
128   ierr = PetscStrcmp(alg,"hypre",&flg);CHKERRQ(ierr);
129   if (flg) {
130     ierr = MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A,B,fill,C);CHKERRQ(ierr);
131     PetscFunctionReturn(0);
132   }
133 #endif
134 
135   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Mat Product Algorithm is not supported");
136 }
137 
138 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(Mat A,Mat B,PetscReal fill,Mat C)
139 {
140   PetscErrorCode     ierr;
141   Mat_SeqAIJ         *a =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
142   PetscInt           *ai=a->i,*bi=b->i,*ci,*cj;
143   PetscInt           am =A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
144   PetscReal          afill;
145   PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,*lnk,ndouble=0,Crmax;
146   PetscTable         ta;
147   PetscBT            lnkbt;
148   PetscFreeSpaceList free_space=NULL,current_space=NULL;
149 
150   PetscFunctionBegin;
151   /* Get ci and cj */
152   /*---------------*/
153   /* Allocate ci array, arrays for fill computation and */
154   /* free space for accumulating nonzero column info */
155   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
156   ci[0] = 0;
157 
158   /* create and initialize a linked list */
159   ierr = PetscTableCreate(bn,bn,&ta);CHKERRQ(ierr);
160   MatRowMergeMax_SeqAIJ(b,bm,ta);
161   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
162   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
163 
164   ierr = PetscLLCondensedCreate(Crmax,bn,&lnk,&lnkbt);CHKERRQ(ierr);
165 
166   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
167   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
168 
169   current_space = free_space;
170 
171   /* Determine ci and cj */
172   for (i=0; i<am; i++) {
173     anzi = ai[i+1] - ai[i];
174     aj   = a->j + ai[i];
175     for (j=0; j<anzi; j++) {
176       brow = aj[j];
177       bnzj = bi[brow+1] - bi[brow];
178       bj   = b->j + bi[brow];
179       /* add non-zero cols of B into the sorted linked list lnk */
180       ierr = PetscLLCondensedAddSorted(bnzj,bj,lnk,lnkbt);CHKERRQ(ierr);
181     }
182     cnzi = lnk[0];
183 
184     /* If free space is not available, make more free space */
185     /* Double the amount of total space in the list */
186     if (current_space->local_remaining<cnzi) {
187       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(cnzi,current_space->total_array_size),&current_space);CHKERRQ(ierr);
188       ndouble++;
189     }
190 
191     /* Copy data into free space, then initialize lnk */
192     ierr = PetscLLCondensedClean(bn,cnzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
193 
194     current_space->array           += cnzi;
195     current_space->local_used      += cnzi;
196     current_space->local_remaining -= cnzi;
197 
198     ci[i+1] = ci[i] + cnzi;
199   }
200 
201   /* Column indices are in the list of free space */
202   /* Allocate space for cj, initialize cj, and */
203   /* destroy list of free space and other temporary array(s) */
204   ierr = PetscMalloc1(ci[am]+1,&cj);CHKERRQ(ierr);
205   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
206   ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
207 
208   /* put together the new symbolic matrix */
209   ierr = MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,((PetscObject)A)->type_name,C);CHKERRQ(ierr);
210   ierr = MatSetBlockSizesFromMats(C,A,B);CHKERRQ(ierr);
211 
212   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
213   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
214   c          = (Mat_SeqAIJ*)(C->data);
215   c->free_a  = PETSC_FALSE;
216   c->free_ij = PETSC_TRUE;
217   c->nonew   = 0;
218 
219   /* fast, needs non-scalable O(bn) array 'abdense' */
220   C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
221 
222   /* set MatInfo */
223   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
224   if (afill < 1.0) afill = 1.0;
225   c->maxnz                  = ci[am];
226   c->nz                     = ci[am];
227   C->info.mallocs           = ndouble;
228   C->info.fill_ratio_given  = fill;
229   C->info.fill_ratio_needed = afill;
230 
231 #if defined(PETSC_USE_INFO)
232   if (ci[am]) {
233     ierr = PetscInfo3(C,"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
234     ierr = PetscInfo1(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
235   } else {
236     ierr = PetscInfo(C,"Empty matrix product\n");CHKERRQ(ierr);
237   }
238 #endif
239   PetscFunctionReturn(0);
240 }
241 
242 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted(Mat A,Mat B,Mat C)
243 {
244   PetscErrorCode ierr;
245   PetscLogDouble flops=0.0;
246   Mat_SeqAIJ     *a   = (Mat_SeqAIJ*)A->data;
247   Mat_SeqAIJ     *b   = (Mat_SeqAIJ*)B->data;
248   Mat_SeqAIJ     *c   = (Mat_SeqAIJ*)C->data;
249   PetscInt       *ai  =a->i,*aj=a->j,*bi=b->i,*bj=b->j,*bjj,*ci=c->i,*cj=c->j;
250   PetscInt       am   =A->rmap->n,cm=C->rmap->n;
251   PetscInt       i,j,k,anzi,bnzi,cnzi,brow;
252   PetscScalar    *aa=a->a,*ba=b->a,*baj,*ca,valtmp;
253   PetscScalar    *ab_dense;
254   PetscContainer cab_dense;
255 
256   PetscFunctionBegin;
257   if (!c->a) { /* first call of MatMatMultNumeric_SeqAIJ_SeqAIJ, allocate ca and matmult_abdense */
258     ierr      = PetscMalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
259     c->a      = ca;
260     c->free_a = PETSC_TRUE;
261   } else ca = c->a;
262 
263   /* TODO this should be done in the symbolic phase */
264   /* However, this function is so heavily used (sometimes in an hidden way through multnumeric function pointers
265      that is hard to eradicate) */
266   ierr = PetscObjectQuery((PetscObject)C,"__PETSc__ab_dense",(PetscObject*)&cab_dense);CHKERRQ(ierr);
267   if (!cab_dense) {
268     ierr = PetscMalloc1(B->cmap->N,&ab_dense);CHKERRQ(ierr);
269     ierr = PetscContainerCreate(PETSC_COMM_SELF,&cab_dense);CHKERRQ(ierr);
270     ierr = PetscContainerSetPointer(cab_dense,ab_dense);CHKERRQ(ierr);
271     ierr = PetscContainerSetUserDestroy(cab_dense,PetscContainerUserDestroyDefault);CHKERRQ(ierr);
272     ierr = PetscObjectCompose((PetscObject)C,"__PETSc__ab_dense",(PetscObject)cab_dense);CHKERRQ(ierr);
273     ierr = PetscObjectDereference((PetscObject)cab_dense);CHKERRQ(ierr);
274   }
275   ierr = PetscContainerGetPointer(cab_dense,(void**)&ab_dense);CHKERRQ(ierr);
276   ierr = PetscArrayzero(ab_dense,B->cmap->N);CHKERRQ(ierr);
277 
278   /* clean old values in C */
279   ierr = PetscArrayzero(ca,ci[cm]);CHKERRQ(ierr);
280   /* Traverse A row-wise. */
281   /* Build the ith row in C by summing over nonzero columns in A, */
282   /* the rows of B corresponding to nonzeros of A. */
283   for (i=0; i<am; i++) {
284     anzi = ai[i+1] - ai[i];
285     for (j=0; j<anzi; j++) {
286       brow = aj[j];
287       bnzi = bi[brow+1] - bi[brow];
288       bjj  = bj + bi[brow];
289       baj  = ba + bi[brow];
290       /* perform dense axpy */
291       valtmp = aa[j];
292       for (k=0; k<bnzi; k++) {
293         ab_dense[bjj[k]] += valtmp*baj[k];
294       }
295       flops += 2*bnzi;
296     }
297     aj += anzi; aa += anzi;
298 
299     cnzi = ci[i+1] - ci[i];
300     for (k=0; k<cnzi; k++) {
301       ca[k]          += ab_dense[cj[k]];
302       ab_dense[cj[k]] = 0.0; /* zero ab_dense */
303     }
304     flops += cnzi;
305     cj    += cnzi; ca += cnzi;
306   }
307   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
308   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
309   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
310   PetscFunctionReturn(0);
311 }
312 
313 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable(Mat A,Mat B,Mat C)
314 {
315   PetscErrorCode ierr;
316   PetscLogDouble flops=0.0;
317   Mat_SeqAIJ     *a   = (Mat_SeqAIJ*)A->data;
318   Mat_SeqAIJ     *b   = (Mat_SeqAIJ*)B->data;
319   Mat_SeqAIJ     *c   = (Mat_SeqAIJ*)C->data;
320   PetscInt       *ai  = a->i,*aj=a->j,*bi=b->i,*bj=b->j,*bjj,*ci=c->i,*cj=c->j;
321   PetscInt       am   = A->rmap->N,cm=C->rmap->N;
322   PetscInt       i,j,k,anzi,bnzi,cnzi,brow;
323   PetscScalar    *aa=a->a,*ba=b->a,*baj,*ca=c->a,valtmp;
324   PetscInt       nextb;
325 
326   PetscFunctionBegin;
327   if (!ca) { /* first call of MatMatMultNumeric_SeqAIJ_SeqAIJ, allocate ca and matmult_abdense */
328     ierr      = PetscMalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
329     c->a      = ca;
330     c->free_a = PETSC_TRUE;
331   }
332 
333   /* clean old values in C */
334   ierr = PetscArrayzero(ca,ci[cm]);CHKERRQ(ierr);
335   /* Traverse A row-wise. */
336   /* Build the ith row in C by summing over nonzero columns in A, */
337   /* the rows of B corresponding to nonzeros of A. */
338   for (i=0; i<am; i++) {
339     anzi = ai[i+1] - ai[i];
340     cnzi = ci[i+1] - ci[i];
341     for (j=0; j<anzi; j++) {
342       brow = aj[j];
343       bnzi = bi[brow+1] - bi[brow];
344       bjj  = bj + bi[brow];
345       baj  = ba + bi[brow];
346       /* perform sparse axpy */
347       valtmp = aa[j];
348       nextb  = 0;
349       for (k=0; nextb<bnzi; k++) {
350         if (cj[k] == bjj[nextb]) { /* ccol == bcol */
351           ca[k] += valtmp*baj[nextb++];
352         }
353       }
354       flops += 2*bnzi;
355     }
356     aj += anzi; aa += anzi;
357     cj += cnzi; ca += cnzi;
358   }
359   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
360   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
361   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
362   PetscFunctionReturn(0);
363 }
364 
365 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable_fast(Mat A,Mat B,PetscReal fill,Mat C)
366 {
367   PetscErrorCode     ierr;
368   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
369   PetscInt           *ai = a->i,*bi=b->i,*ci,*cj;
370   PetscInt           am  = A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
371   MatScalar          *ca;
372   PetscReal          afill;
373   PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,*lnk,ndouble=0,Crmax;
374   PetscTable         ta;
375   PetscFreeSpaceList free_space=NULL,current_space=NULL;
376 
377   PetscFunctionBegin;
378   /* Get ci and cj - same as MatMatMultSymbolic_SeqAIJ_SeqAIJ except using PetscLLxxx_fast() */
379   /*-----------------------------------------------------------------------------------------*/
380   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
381   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
382   ci[0] = 0;
383 
384   /* create and initialize a linked list */
385   ierr = PetscTableCreate(bn,bn,&ta);CHKERRQ(ierr);
386   MatRowMergeMax_SeqAIJ(b,bm,ta);
387   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
388   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
389 
390   ierr = PetscLLCondensedCreate_fast(Crmax,&lnk);CHKERRQ(ierr);
391 
392   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
393   ierr          = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
394   current_space = free_space;
395 
396   /* Determine ci and cj */
397   for (i=0; i<am; i++) {
398     anzi = ai[i+1] - ai[i];
399     aj   = a->j + ai[i];
400     for (j=0; j<anzi; j++) {
401       brow = aj[j];
402       bnzj = bi[brow+1] - bi[brow];
403       bj   = b->j + bi[brow];
404       /* add non-zero cols of B into the sorted linked list lnk */
405       ierr = PetscLLCondensedAddSorted_fast(bnzj,bj,lnk);CHKERRQ(ierr);
406     }
407     cnzi = lnk[1];
408 
409     /* If free space is not available, make more free space */
410     /* Double the amount of total space in the list */
411     if (current_space->local_remaining<cnzi) {
412       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(cnzi,current_space->total_array_size),&current_space);CHKERRQ(ierr);
413       ndouble++;
414     }
415 
416     /* Copy data into free space, then initialize lnk */
417     ierr = PetscLLCondensedClean_fast(cnzi,current_space->array,lnk);CHKERRQ(ierr);
418 
419     current_space->array           += cnzi;
420     current_space->local_used      += cnzi;
421     current_space->local_remaining -= cnzi;
422 
423     ci[i+1] = ci[i] + cnzi;
424   }
425 
426   /* Column indices are in the list of free space */
427   /* Allocate space for cj, initialize cj, and */
428   /* destroy list of free space and other temporary array(s) */
429   ierr = PetscMalloc1(ci[am]+1,&cj);CHKERRQ(ierr);
430   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
431   ierr = PetscLLCondensedDestroy_fast(lnk);CHKERRQ(ierr);
432 
433   /* Allocate space for ca */
434   ierr = PetscCalloc1(ci[am]+1,&ca);CHKERRQ(ierr);
435 
436   /* put together the new symbolic matrix */
437   ierr = MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A),am,bn,ci,cj,ca,((PetscObject)A)->type_name,C);CHKERRQ(ierr);
438   ierr = MatSetBlockSizesFromMats(C,A,B);CHKERRQ(ierr);
439 
440   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
441   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
442   c          = (Mat_SeqAIJ*)(C->data);
443   c->free_a  = PETSC_TRUE;
444   c->free_ij = PETSC_TRUE;
445   c->nonew   = 0;
446 
447   /* slower, less memory */
448   C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable;
449 
450   /* set MatInfo */
451   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
452   if (afill < 1.0) afill = 1.0;
453   c->maxnz                     = ci[am];
454   c->nz                        = ci[am];
455   C->info.mallocs           = ndouble;
456   C->info.fill_ratio_given  = fill;
457   C->info.fill_ratio_needed = afill;
458 
459 #if defined(PETSC_USE_INFO)
460   if (ci[am]) {
461     ierr = PetscInfo3(C,"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
462     ierr = PetscInfo1(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
463   } else {
464     ierr = PetscInfo(C,"Empty matrix product\n");CHKERRQ(ierr);
465   }
466 #endif
467   PetscFunctionReturn(0);
468 }
469 
470 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable(Mat A,Mat B,PetscReal fill,Mat C)
471 {
472   PetscErrorCode     ierr;
473   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
474   PetscInt           *ai = a->i,*bi=b->i,*ci,*cj;
475   PetscInt           am  = A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
476   MatScalar          *ca;
477   PetscReal          afill;
478   PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,*lnk,ndouble=0,Crmax;
479   PetscTable         ta;
480   PetscFreeSpaceList free_space=NULL,current_space=NULL;
481 
482   PetscFunctionBegin;
483   /* Get ci and cj - same as MatMatMultSymbolic_SeqAIJ_SeqAIJ except using PetscLLxxx_Scalalbe() */
484   /*---------------------------------------------------------------------------------------------*/
485   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
486   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
487   ci[0] = 0;
488 
489   /* create and initialize a linked list */
490   ierr = PetscTableCreate(bn,bn,&ta);CHKERRQ(ierr);
491   MatRowMergeMax_SeqAIJ(b,bm,ta);
492   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
493   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
494   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
495 
496   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
497   ierr          = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
498   current_space = free_space;
499 
500   /* Determine ci and cj */
501   for (i=0; i<am; i++) {
502     anzi = ai[i+1] - ai[i];
503     aj   = a->j + ai[i];
504     for (j=0; j<anzi; j++) {
505       brow = aj[j];
506       bnzj = bi[brow+1] - bi[brow];
507       bj   = b->j + bi[brow];
508       /* add non-zero cols of B into the sorted linked list lnk */
509       ierr = PetscLLCondensedAddSorted_Scalable(bnzj,bj,lnk);CHKERRQ(ierr);
510     }
511     cnzi = lnk[0];
512 
513     /* If free space is not available, make more free space */
514     /* Double the amount of total space in the list */
515     if (current_space->local_remaining<cnzi) {
516       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(cnzi,current_space->total_array_size),&current_space);CHKERRQ(ierr);
517       ndouble++;
518     }
519 
520     /* Copy data into free space, then initialize lnk */
521     ierr = PetscLLCondensedClean_Scalable(cnzi,current_space->array,lnk);CHKERRQ(ierr);
522 
523     current_space->array           += cnzi;
524     current_space->local_used      += cnzi;
525     current_space->local_remaining -= cnzi;
526 
527     ci[i+1] = ci[i] + cnzi;
528   }
529 
530   /* Column indices are in the list of free space */
531   /* Allocate space for cj, initialize cj, and */
532   /* destroy list of free space and other temporary array(s) */
533   ierr = PetscMalloc1(ci[am]+1,&cj);CHKERRQ(ierr);
534   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
535   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
536 
537   /* Allocate space for ca */
538   /*-----------------------*/
539   ierr = PetscCalloc1(ci[am]+1,&ca);CHKERRQ(ierr);
540 
541   /* put together the new symbolic matrix */
542   ierr = MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A),am,bn,ci,cj,ca,((PetscObject)A)->type_name,C);CHKERRQ(ierr);
543   ierr = MatSetBlockSizesFromMats(C,A,B);CHKERRQ(ierr);
544 
545   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
546   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
547   c          = (Mat_SeqAIJ*)(C->data);
548   c->free_a  = PETSC_TRUE;
549   c->free_ij = PETSC_TRUE;
550   c->nonew   = 0;
551 
552   /* slower, less memory */
553   C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable;
554 
555   /* set MatInfo */
556   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
557   if (afill < 1.0) afill = 1.0;
558   c->maxnz                     = ci[am];
559   c->nz                        = ci[am];
560   C->info.mallocs           = ndouble;
561   C->info.fill_ratio_given  = fill;
562   C->info.fill_ratio_needed = afill;
563 
564 #if defined(PETSC_USE_INFO)
565   if (ci[am]) {
566     ierr = PetscInfo3(C,"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
567     ierr = PetscInfo1(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
568   } else {
569     ierr = PetscInfo(C,"Empty matrix product\n");CHKERRQ(ierr);
570   }
571 #endif
572   PetscFunctionReturn(0);
573 }
574 
575 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Heap(Mat A,Mat B,PetscReal fill,Mat C)
576 {
577   PetscErrorCode     ierr;
578   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
579   const PetscInt     *ai=a->i,*bi=b->i,*aj=a->j,*bj=b->j;
580   PetscInt           *ci,*cj,*bb;
581   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
582   PetscReal          afill;
583   PetscInt           i,j,col,ndouble = 0;
584   PetscFreeSpaceList free_space=NULL,current_space=NULL;
585   PetscHeap          h;
586 
587   PetscFunctionBegin;
588   /* Get ci and cj - by merging sorted rows using a heap */
589   /*---------------------------------------------------------------------------------------------*/
590   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
591   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
592   ci[0] = 0;
593 
594   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
595   ierr          = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
596   current_space = free_space;
597 
598   ierr = PetscHeapCreate(a->rmax,&h);CHKERRQ(ierr);
599   ierr = PetscMalloc1(a->rmax,&bb);CHKERRQ(ierr);
600 
601   /* Determine ci and cj */
602   for (i=0; i<am; i++) {
603     const PetscInt anzi  = ai[i+1] - ai[i]; /* number of nonzeros in this row of A, this is the number of rows of B that we merge */
604     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
605     ci[i+1] = ci[i];
606     /* Populate the min heap */
607     for (j=0; j<anzi; j++) {
608       bb[j] = bi[acol[j]];         /* bb points at the start of the row */
609       if (bb[j] < bi[acol[j]+1]) { /* Add if row is nonempty */
610         ierr = PetscHeapAdd(h,j,bj[bb[j]++]);CHKERRQ(ierr);
611       }
612     }
613     /* Pick off the min element, adding it to free space */
614     ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
615     while (j >= 0) {
616       if (current_space->local_remaining < 1) { /* double the size, but don't exceed 16 MiB */
617         ierr = PetscFreeSpaceGet(PetscMin(PetscIntMultTruncate(2,current_space->total_array_size),16 << 20),&current_space);CHKERRQ(ierr);
618         ndouble++;
619       }
620       *(current_space->array++) = col;
621       current_space->local_used++;
622       current_space->local_remaining--;
623       ci[i+1]++;
624 
625       /* stash if anything else remains in this row of B */
626       if (bb[j] < bi[acol[j]+1]) {ierr = PetscHeapStash(h,j,bj[bb[j]++]);CHKERRQ(ierr);}
627       while (1) {               /* pop and stash any other rows of B that also had an entry in this column */
628         PetscInt j2,col2;
629         ierr = PetscHeapPeek(h,&j2,&col2);CHKERRQ(ierr);
630         if (col2 != col) break;
631         ierr = PetscHeapPop(h,&j2,&col2);CHKERRQ(ierr);
632         if (bb[j2] < bi[acol[j2]+1]) {ierr = PetscHeapStash(h,j2,bj[bb[j2]++]);CHKERRQ(ierr);}
633       }
634       /* Put any stashed elements back into the min heap */
635       ierr = PetscHeapUnstash(h);CHKERRQ(ierr);
636       ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
637     }
638   }
639   ierr = PetscFree(bb);CHKERRQ(ierr);
640   ierr = PetscHeapDestroy(&h);CHKERRQ(ierr);
641 
642   /* Column indices are in the list of free space */
643   /* Allocate space for cj, initialize cj, and */
644   /* destroy list of free space and other temporary array(s) */
645   ierr = PetscMalloc1(ci[am],&cj);CHKERRQ(ierr);
646   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
647 
648   /* put together the new symbolic matrix */
649   ierr = MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,((PetscObject)A)->type_name,C);CHKERRQ(ierr);
650   ierr = MatSetBlockSizesFromMats(C,A,B);CHKERRQ(ierr);
651 
652   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
653   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
654   c          = (Mat_SeqAIJ*)(C->data);
655   c->free_a  = PETSC_TRUE;
656   c->free_ij = PETSC_TRUE;
657   c->nonew   = 0;
658 
659   C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
660 
661   /* set MatInfo */
662   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
663   if (afill < 1.0) afill = 1.0;
664   c->maxnz                     = ci[am];
665   c->nz                        = ci[am];
666   C->info.mallocs           = ndouble;
667   C->info.fill_ratio_given  = fill;
668   C->info.fill_ratio_needed = afill;
669 
670 #if defined(PETSC_USE_INFO)
671   if (ci[am]) {
672     ierr = PetscInfo3(C,"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
673     ierr = PetscInfo1(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
674   } else {
675     ierr = PetscInfo(C,"Empty matrix product\n");CHKERRQ(ierr);
676   }
677 #endif
678   PetscFunctionReturn(0);
679 }
680 
681 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_BTHeap(Mat A,Mat B,PetscReal fill,Mat C)
682 {
683   PetscErrorCode     ierr;
684   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
685   const PetscInt     *ai = a->i,*bi=b->i,*aj=a->j,*bj=b->j;
686   PetscInt           *ci,*cj,*bb;
687   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
688   PetscReal          afill;
689   PetscInt           i,j,col,ndouble = 0;
690   PetscFreeSpaceList free_space=NULL,current_space=NULL;
691   PetscHeap          h;
692   PetscBT            bt;
693 
694   PetscFunctionBegin;
695   /* Get ci and cj - using a heap for the sorted rows, but use BT so that each index is only added once */
696   /*---------------------------------------------------------------------------------------------*/
697   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
698   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
699   ci[0] = 0;
700 
701   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
702   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
703 
704   current_space = free_space;
705 
706   ierr = PetscHeapCreate(a->rmax,&h);CHKERRQ(ierr);
707   ierr = PetscMalloc1(a->rmax,&bb);CHKERRQ(ierr);
708   ierr = PetscBTCreate(bn,&bt);CHKERRQ(ierr);
709 
710   /* Determine ci and cj */
711   for (i=0; i<am; i++) {
712     const PetscInt anzi  = ai[i+1] - ai[i]; /* number of nonzeros in this row of A, this is the number of rows of B that we merge */
713     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
714     const PetscInt *fptr = current_space->array; /* Save beginning of the row so we can clear the BT later */
715     ci[i+1] = ci[i];
716     /* Populate the min heap */
717     for (j=0; j<anzi; j++) {
718       PetscInt brow = acol[j];
719       for (bb[j] = bi[brow]; bb[j] < bi[brow+1]; bb[j]++) {
720         PetscInt bcol = bj[bb[j]];
721         if (!PetscBTLookupSet(bt,bcol)) { /* new entry */
722           ierr = PetscHeapAdd(h,j,bcol);CHKERRQ(ierr);
723           bb[j]++;
724           break;
725         }
726       }
727     }
728     /* Pick off the min element, adding it to free space */
729     ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
730     while (j >= 0) {
731       if (current_space->local_remaining < 1) { /* double the size, but don't exceed 16 MiB */
732         fptr = NULL;                      /* need PetscBTMemzero */
733         ierr = PetscFreeSpaceGet(PetscMin(PetscIntMultTruncate(2,current_space->total_array_size),16 << 20),&current_space);CHKERRQ(ierr);
734         ndouble++;
735       }
736       *(current_space->array++) = col;
737       current_space->local_used++;
738       current_space->local_remaining--;
739       ci[i+1]++;
740 
741       /* stash if anything else remains in this row of B */
742       for (; bb[j] < bi[acol[j]+1]; bb[j]++) {
743         PetscInt bcol = bj[bb[j]];
744         if (!PetscBTLookupSet(bt,bcol)) { /* new entry */
745           ierr = PetscHeapAdd(h,j,bcol);CHKERRQ(ierr);
746           bb[j]++;
747           break;
748         }
749       }
750       ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
751     }
752     if (fptr) {                 /* Clear the bits for this row */
753       for (; fptr<current_space->array; fptr++) {ierr = PetscBTClear(bt,*fptr);CHKERRQ(ierr);}
754     } else {                    /* We reallocated so we don't remember (easily) how to clear only the bits we changed */
755       ierr = PetscBTMemzero(bn,bt);CHKERRQ(ierr);
756     }
757   }
758   ierr = PetscFree(bb);CHKERRQ(ierr);
759   ierr = PetscHeapDestroy(&h);CHKERRQ(ierr);
760   ierr = PetscBTDestroy(&bt);CHKERRQ(ierr);
761 
762   /* Column indices are in the list of free space */
763   /* Allocate space for cj, initialize cj, and */
764   /* destroy list of free space and other temporary array(s) */
765   ierr = PetscMalloc1(ci[am],&cj);CHKERRQ(ierr);
766   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
767 
768   /* put together the new symbolic matrix */
769   ierr = MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,((PetscObject)A)->type_name,C);CHKERRQ(ierr);
770   ierr = MatSetBlockSizesFromMats(C,A,B);CHKERRQ(ierr);
771 
772   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
773   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
774   c          = (Mat_SeqAIJ*)(C->data);
775   c->free_a  = PETSC_TRUE;
776   c->free_ij = PETSC_TRUE;
777   c->nonew   = 0;
778 
779   C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
780 
781   /* set MatInfo */
782   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
783   if (afill < 1.0) afill = 1.0;
784   c->maxnz                     = ci[am];
785   c->nz                        = ci[am];
786   C->info.mallocs           = ndouble;
787   C->info.fill_ratio_given  = fill;
788   C->info.fill_ratio_needed = afill;
789 
790 #if defined(PETSC_USE_INFO)
791   if (ci[am]) {
792     ierr = PetscInfo3(C,"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
793     ierr = PetscInfo1(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
794   } else {
795     ierr = PetscInfo(C,"Empty matrix product\n");CHKERRQ(ierr);
796   }
797 #endif
798   PetscFunctionReturn(0);
799 }
800 
801 
802 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_RowMerge(Mat A,Mat B,PetscReal fill,Mat C)
803 {
804   PetscErrorCode     ierr;
805   Mat_SeqAIJ         *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
806   const PetscInt     *ai=a->i,*bi=b->i,*aj=a->j,*bj=b->j,*inputi,*inputj,*inputcol,*inputcol_L1;
807   PetscInt           *ci,*cj,*outputj,worki_L1[9],worki_L2[9];
808   PetscInt           c_maxmem,a_maxrownnz=0,a_rownnz;
809   const PetscInt     workcol[8]={0,1,2,3,4,5,6,7};
810   const PetscInt     am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
811   const PetscInt     *brow_ptr[8],*brow_end[8];
812   PetscInt           window[8];
813   PetscInt           window_min,old_window_min,ci_nnz,outputi_nnz=0,L1_nrows,L2_nrows;
814   PetscInt           i,k,ndouble=0,L1_rowsleft,rowsleft;
815   PetscReal          afill;
816   PetscInt           *workj_L1,*workj_L2,*workj_L3;
817   PetscInt           L1_nnz,L2_nnz;
818 
819   /* Step 1: Get upper bound on memory required for allocation.
820              Because of the way virtual memory works,
821              only the memory pages that are actually needed will be physically allocated. */
822   PetscFunctionBegin;
823   ierr = PetscMalloc1(am+1,&ci);CHKERRQ(ierr);
824   for (i=0; i<am; i++) {
825     const PetscInt anzi  = ai[i+1] - ai[i]; /* number of nonzeros in this row of A, this is the number of rows of B that we merge */
826     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
827     a_rownnz = 0;
828     for (k=0; k<anzi; ++k) {
829       a_rownnz += bi[acol[k]+1] - bi[acol[k]];
830       if (a_rownnz > bn) {
831         a_rownnz = bn;
832         break;
833       }
834     }
835     a_maxrownnz = PetscMax(a_maxrownnz, a_rownnz);
836   }
837   /* temporary work areas for merging rows */
838   ierr = PetscMalloc1(a_maxrownnz*8,&workj_L1);CHKERRQ(ierr);
839   ierr = PetscMalloc1(a_maxrownnz*8,&workj_L2);CHKERRQ(ierr);
840   ierr = PetscMalloc1(a_maxrownnz,&workj_L3);CHKERRQ(ierr);
841 
842   /* This should be enough for almost all matrices. If not, memory is reallocated later. */
843   c_maxmem = 8*(ai[am]+bi[bm]);
844   /* Step 2: Populate pattern for C */
845   ierr  = PetscMalloc1(c_maxmem,&cj);CHKERRQ(ierr);
846 
847   ci_nnz       = 0;
848   ci[0]        = 0;
849   worki_L1[0]  = 0;
850   worki_L2[0]  = 0;
851   for (i=0; i<am; i++) {
852     const PetscInt anzi  = ai[i+1] - ai[i]; /* number of nonzeros in this row of A, this is the number of rows of B that we merge */
853     const PetscInt *acol = aj + ai[i];      /* column indices of nonzero entries in this row */
854     rowsleft             = anzi;
855     inputcol_L1          = acol;
856     L2_nnz               = 0;
857     L2_nrows             = 1;  /* Number of rows to be merged on Level 3. output of L3 already exists -> initial value 1   */
858     worki_L2[1]          = 0;
859     outputi_nnz          = 0;
860 
861     /* If the number of indices in C so far + the max number of columns in the next row > c_maxmem  -> allocate more memory */
862     while (ci_nnz+a_maxrownnz > c_maxmem) {
863       c_maxmem *= 2;
864       ndouble++;
865       ierr = PetscRealloc(sizeof(PetscInt)*c_maxmem,&cj);CHKERRQ(ierr);
866     }
867 
868     while (rowsleft) {
869       L1_rowsleft = PetscMin(64, rowsleft); /* In the inner loop max 64 rows of B can be merged */
870       L1_nrows    = 0;
871       L1_nnz      = 0;
872       inputcol    = inputcol_L1;
873       inputi      = bi;
874       inputj      = bj;
875 
876       /* The following macro is used to specialize for small rows in A.
877          This helps with compiler unrolling, improving performance substantially.
878           Input:  inputj   inputi  inputcol  bn
879           Output: outputj  outputi_nnz                       */
880        #define MatMatMultSymbolic_RowMergeMacro(ANNZ)                        \
881          window_min  = bn;                                                   \
882          outputi_nnz = 0;                                                    \
883          for (k=0; k<ANNZ; ++k) {                                            \
884            brow_ptr[k] = inputj + inputi[inputcol[k]];                       \
885            brow_end[k] = inputj + inputi[inputcol[k]+1];                     \
886            window[k]   = (brow_ptr[k] != brow_end[k]) ? *brow_ptr[k] : bn;   \
887            window_min  = PetscMin(window[k], window_min);                    \
888          }                                                                   \
889          while (window_min < bn) {                                           \
890            outputj[outputi_nnz++] = window_min;                              \
891            /* advance front and compute new minimum */                       \
892            old_window_min = window_min;                                      \
893            window_min = bn;                                                  \
894            for (k=0; k<ANNZ; ++k) {                                          \
895              if (window[k] == old_window_min) {                              \
896                brow_ptr[k]++;                                                \
897                window[k] = (brow_ptr[k] != brow_end[k]) ? *brow_ptr[k] : bn; \
898              }                                                               \
899              window_min = PetscMin(window[k], window_min);                   \
900            }                                                                 \
901          }
902 
903       /************** L E V E L  1 ***************/
904       /* Merge up to 8 rows of B to L1 work array*/
905       while (L1_rowsleft) {
906         outputi_nnz = 0;
907         if (anzi > 8)  outputj = workj_L1 + L1_nnz;     /* Level 1 rowmerge*/
908         else           outputj = cj + ci_nnz;           /* Merge directly to C */
909 
910         switch (L1_rowsleft) {
911         case 1:  brow_ptr[0] = inputj + inputi[inputcol[0]];
912                  brow_end[0] = inputj + inputi[inputcol[0]+1];
913                  for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
914                  inputcol    += L1_rowsleft;
915                  rowsleft    -= L1_rowsleft;
916                  L1_rowsleft  = 0;
917                  break;
918         case 2:  MatMatMultSymbolic_RowMergeMacro(2);
919                  inputcol    += L1_rowsleft;
920                  rowsleft    -= L1_rowsleft;
921                  L1_rowsleft  = 0;
922                  break;
923         case 3: MatMatMultSymbolic_RowMergeMacro(3);
924                  inputcol    += L1_rowsleft;
925                  rowsleft    -= L1_rowsleft;
926                  L1_rowsleft  = 0;
927                  break;
928         case 4:  MatMatMultSymbolic_RowMergeMacro(4);
929                  inputcol    += L1_rowsleft;
930                  rowsleft    -= L1_rowsleft;
931                  L1_rowsleft  = 0;
932                  break;
933         case 5:  MatMatMultSymbolic_RowMergeMacro(5);
934                  inputcol    += L1_rowsleft;
935                  rowsleft    -= L1_rowsleft;
936                  L1_rowsleft  = 0;
937                  break;
938         case 6:  MatMatMultSymbolic_RowMergeMacro(6);
939                  inputcol    += L1_rowsleft;
940                  rowsleft    -= L1_rowsleft;
941                  L1_rowsleft  = 0;
942                  break;
943         case 7:  MatMatMultSymbolic_RowMergeMacro(7);
944                  inputcol    += L1_rowsleft;
945                  rowsleft    -= L1_rowsleft;
946                  L1_rowsleft  = 0;
947                  break;
948         default: MatMatMultSymbolic_RowMergeMacro(8);
949                  inputcol    += 8;
950                  rowsleft    -= 8;
951                  L1_rowsleft -= 8;
952                  break;
953         }
954         inputcol_L1           = inputcol;
955         L1_nnz               += outputi_nnz;
956         worki_L1[++L1_nrows]  = L1_nnz;
957       }
958 
959       /********************** L E V E L  2 ************************/
960       /* Merge from L1 work array to either C or to L2 work array */
961       if (anzi > 8) {
962         inputi      = worki_L1;
963         inputj      = workj_L1;
964         inputcol    = workcol;
965         outputi_nnz = 0;
966 
967         if (anzi <= 64) outputj = cj + ci_nnz;        /* Merge from L1 work array to C */
968         else            outputj = workj_L2 + L2_nnz;  /* Merge from L1 work array to L2 work array */
969 
970         switch (L1_nrows) {
971         case 1:  brow_ptr[0] = inputj + inputi[inputcol[0]];
972                  brow_end[0] = inputj + inputi[inputcol[0]+1];
973                  for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
974                  break;
975         case 2:  MatMatMultSymbolic_RowMergeMacro(2); break;
976         case 3:  MatMatMultSymbolic_RowMergeMacro(3); break;
977         case 4:  MatMatMultSymbolic_RowMergeMacro(4); break;
978         case 5:  MatMatMultSymbolic_RowMergeMacro(5); break;
979         case 6:  MatMatMultSymbolic_RowMergeMacro(6); break;
980         case 7:  MatMatMultSymbolic_RowMergeMacro(7); break;
981         case 8:  MatMatMultSymbolic_RowMergeMacro(8); break;
982         default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MatMatMult logic error: Not merging 1-8 rows from L1 work array!");
983         }
984         L2_nnz               += outputi_nnz;
985         worki_L2[++L2_nrows]  = L2_nnz;
986 
987         /************************ L E V E L  3 **********************/
988         /* Merge from L2 work array to either C or to L2 work array */
989         if (anzi > 64 && (L2_nrows == 8 || rowsleft == 0)) {
990           inputi      = worki_L2;
991           inputj      = workj_L2;
992           inputcol    = workcol;
993           outputi_nnz = 0;
994           if (rowsleft) outputj = workj_L3;
995           else          outputj = cj + ci_nnz;
996           switch (L2_nrows) {
997           case 1:  brow_ptr[0] = inputj + inputi[inputcol[0]];
998                    brow_end[0] = inputj + inputi[inputcol[0]+1];
999                    for (; brow_ptr[0] != brow_end[0]; ++brow_ptr[0]) outputj[outputi_nnz++] = *brow_ptr[0]; /* copy row in b over */
1000                    break;
1001           case 2:  MatMatMultSymbolic_RowMergeMacro(2); break;
1002           case 3:  MatMatMultSymbolic_RowMergeMacro(3); break;
1003           case 4:  MatMatMultSymbolic_RowMergeMacro(4); break;
1004           case 5:  MatMatMultSymbolic_RowMergeMacro(5); break;
1005           case 6:  MatMatMultSymbolic_RowMergeMacro(6); break;
1006           case 7:  MatMatMultSymbolic_RowMergeMacro(7); break;
1007           case 8:  MatMatMultSymbolic_RowMergeMacro(8); break;
1008           default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MatMatMult logic error: Not merging 1-8 rows from L2 work array!");
1009           }
1010           L2_nrows    = 1;
1011           L2_nnz      = outputi_nnz;
1012           worki_L2[1] = outputi_nnz;
1013           /* Copy to workj_L2 */
1014           if (rowsleft) {
1015             for (k=0; k<outputi_nnz; ++k)  workj_L2[k] = outputj[k];
1016           }
1017         }
1018       }
1019     }  /* while (rowsleft) */
1020 #undef MatMatMultSymbolic_RowMergeMacro
1021 
1022     /* terminate current row */
1023     ci_nnz += outputi_nnz;
1024     ci[i+1] = ci_nnz;
1025   }
1026 
1027   /* Step 3: Create the new symbolic matrix */
1028   ierr = MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,((PetscObject)A)->type_name,C);CHKERRQ(ierr);
1029   ierr = MatSetBlockSizesFromMats(C,A,B);CHKERRQ(ierr);
1030 
1031   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
1032   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
1033   c          = (Mat_SeqAIJ*)(C->data);
1034   c->free_a  = PETSC_TRUE;
1035   c->free_ij = PETSC_TRUE;
1036   c->nonew   = 0;
1037 
1038   C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
1039 
1040   /* set MatInfo */
1041   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
1042   if (afill < 1.0) afill = 1.0;
1043   c->maxnz                     = ci[am];
1044   c->nz                        = ci[am];
1045   C->info.mallocs           = ndouble;
1046   C->info.fill_ratio_given  = fill;
1047   C->info.fill_ratio_needed = afill;
1048 
1049 #if defined(PETSC_USE_INFO)
1050   if (ci[am]) {
1051     ierr = PetscInfo3(C,"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
1052     ierr = PetscInfo1(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
1053   } else {
1054     ierr = PetscInfo(C,"Empty matrix product\n");CHKERRQ(ierr);
1055   }
1056 #endif
1057 
1058   /* Step 4: Free temporary work areas */
1059   ierr = PetscFree(workj_L1);CHKERRQ(ierr);
1060   ierr = PetscFree(workj_L2);CHKERRQ(ierr);
1061   ierr = PetscFree(workj_L3);CHKERRQ(ierr);
1062   PetscFunctionReturn(0);
1063 }
1064 
1065 /* concatenate unique entries and then sort */
1066 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Sorted(Mat A,Mat B,PetscReal fill,Mat C)
1067 {
1068   PetscErrorCode ierr;
1069   Mat_SeqAIJ     *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
1070   const PetscInt *ai = a->i,*bi=b->i,*aj=a->j,*bj=b->j;
1071   PetscInt       *ci,*cj;
1072   PetscInt       am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
1073   PetscReal      afill;
1074   PetscInt       i,j,ndouble = 0;
1075   PetscSegBuffer seg,segrow;
1076   char           *seen;
1077 
1078   PetscFunctionBegin;
1079   ierr  = PetscMalloc1(am+1,&ci);CHKERRQ(ierr);
1080   ci[0] = 0;
1081 
1082   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
1083   ierr = PetscSegBufferCreate(sizeof(PetscInt),(PetscInt)(fill*(ai[am]+bi[bm])),&seg);CHKERRQ(ierr);
1084   ierr = PetscSegBufferCreate(sizeof(PetscInt),100,&segrow);CHKERRQ(ierr);
1085   ierr = PetscCalloc1(bn,&seen);CHKERRQ(ierr);
1086 
1087   /* Determine ci and cj */
1088   for (i=0; i<am; i++) {
1089     const PetscInt anzi  = ai[i+1] - ai[i]; /* number of nonzeros in this row of A, this is the number of rows of B that we merge */
1090     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
1091     PetscInt packlen = 0,*PETSC_RESTRICT crow;
1092     /* Pack segrow */
1093     for (j=0; j<anzi; j++) {
1094       PetscInt brow = acol[j],bjstart = bi[brow],bjend = bi[brow+1],k;
1095       for (k=bjstart; k<bjend; k++) {
1096         PetscInt bcol = bj[k];
1097         if (!seen[bcol]) { /* new entry */
1098           PetscInt *PETSC_RESTRICT slot;
1099           ierr = PetscSegBufferGetInts(segrow,1,&slot);CHKERRQ(ierr);
1100           *slot = bcol;
1101           seen[bcol] = 1;
1102           packlen++;
1103         }
1104       }
1105     }
1106     ierr = PetscSegBufferGetInts(seg,packlen,&crow);CHKERRQ(ierr);
1107     ierr = PetscSegBufferExtractTo(segrow,crow);CHKERRQ(ierr);
1108     ierr = PetscSortInt(packlen,crow);CHKERRQ(ierr);
1109     ci[i+1] = ci[i] + packlen;
1110     for (j=0; j<packlen; j++) seen[crow[j]] = 0;
1111   }
1112   ierr = PetscSegBufferDestroy(&segrow);CHKERRQ(ierr);
1113   ierr = PetscFree(seen);CHKERRQ(ierr);
1114 
1115   /* Column indices are in the segmented buffer */
1116   ierr = PetscSegBufferExtractAlloc(seg,&cj);CHKERRQ(ierr);
1117   ierr = PetscSegBufferDestroy(&seg);CHKERRQ(ierr);
1118 
1119   /* put together the new symbolic matrix */
1120   ierr = MatSetSeqAIJWithArrays_private(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,((PetscObject)A)->type_name,C);CHKERRQ(ierr);
1121   ierr = MatSetBlockSizesFromMats(C,A,B);CHKERRQ(ierr);
1122 
1123   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
1124   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
1125   c          = (Mat_SeqAIJ*)(C->data);
1126   c->free_a  = PETSC_TRUE;
1127   c->free_ij = PETSC_TRUE;
1128   c->nonew   = 0;
1129 
1130   C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Sorted;
1131 
1132   /* set MatInfo */
1133   afill = (PetscReal)ci[am]/PetscMax(ai[am]+bi[bm],1) + 1.e-5;
1134   if (afill < 1.0) afill = 1.0;
1135   c->maxnz                  = ci[am];
1136   c->nz                     = ci[am];
1137   C->info.mallocs           = ndouble;
1138   C->info.fill_ratio_given  = fill;
1139   C->info.fill_ratio_needed = afill;
1140 
1141 #if defined(PETSC_USE_INFO)
1142   if (ci[am]) {
1143     ierr = PetscInfo3(C,"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
1144     ierr = PetscInfo1(C,"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
1145   } else {
1146     ierr = PetscInfo(C,"Empty matrix product\n");CHKERRQ(ierr);
1147   }
1148 #endif
1149   PetscFunctionReturn(0);
1150 }
1151 
1152 PetscErrorCode MatDestroy_SeqAIJ_MatMatMultTrans(void *data)
1153 {
1154   PetscErrorCode      ierr;
1155   Mat_MatMatTransMult *abt=(Mat_MatMatTransMult *)data;
1156 
1157   PetscFunctionBegin;
1158   ierr = MatTransposeColoringDestroy(&abt->matcoloring);CHKERRQ(ierr);
1159   ierr = MatDestroy(&abt->Bt_den);CHKERRQ(ierr);
1160   ierr = MatDestroy(&abt->ABt_den);CHKERRQ(ierr);
1161   ierr = PetscFree(abt);CHKERRQ(ierr);
1162   PetscFunctionReturn(0);
1163 }
1164 
1165 PetscErrorCode MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat C)
1166 {
1167   PetscErrorCode      ierr;
1168   Mat                 Bt;
1169   PetscInt            *bti,*btj;
1170   Mat_MatMatTransMult *abt;
1171   Mat_Product         *product = C->product;
1172   char                *alg;
1173 
1174   PetscFunctionBegin;
1175   if (!product) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Missing product struct");
1176   if (product->data) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Extra product struct not empty");
1177 
1178   /* create symbolic Bt */
1179   ierr = MatGetSymbolicTranspose_SeqAIJ(B,&bti,&btj);CHKERRQ(ierr);
1180   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,B->cmap->n,B->rmap->n,bti,btj,NULL,&Bt);CHKERRQ(ierr);
1181   ierr = MatSetBlockSizes(Bt,PetscAbs(A->cmap->bs),PetscAbs(B->cmap->bs));CHKERRQ(ierr);
1182   ierr = MatSetType(Bt,((PetscObject)A)->type_name);CHKERRQ(ierr);
1183 
1184   /* get symbolic C=A*Bt */
1185   ierr = PetscStrallocpy(product->alg,&alg);CHKERRQ(ierr);
1186   ierr = MatProductSetAlgorithm(C,"sorted");CHKERRQ(ierr); /* set algorithm for C = A*Bt */
1187   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(A,Bt,fill,C);CHKERRQ(ierr);
1188   ierr = MatProductSetAlgorithm(C,alg);CHKERRQ(ierr); /* resume original algorithm for ABt product */
1189   ierr = PetscFree(alg);CHKERRQ(ierr);
1190 
1191   /* create a supporting struct for reuse intermidiate dense matrices with matcoloring */
1192   ierr = PetscNew(&abt);CHKERRQ(ierr);
1193 
1194   product->data    = abt;
1195   product->destroy = MatDestroy_SeqAIJ_MatMatMultTrans;
1196 
1197   C->ops->mattransposemultnumeric = MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ;
1198 
1199   abt->usecoloring = PETSC_FALSE;
1200   ierr = PetscStrcmp(product->alg,"color",&abt->usecoloring);CHKERRQ(ierr);
1201   if (abt->usecoloring) {
1202     /* Create MatTransposeColoring from symbolic C=A*B^T */
1203     MatTransposeColoring matcoloring;
1204     MatColoring          coloring;
1205     ISColoring           iscoloring;
1206     Mat                  Bt_dense,C_dense;
1207 
1208     /* inode causes memory problem */
1209     ierr = MatSetOption(C,MAT_USE_INODES,PETSC_FALSE);CHKERRQ(ierr);
1210 
1211     ierr = MatColoringCreate(C,&coloring);CHKERRQ(ierr);
1212     ierr = MatColoringSetDistance(coloring,2);CHKERRQ(ierr);
1213     ierr = MatColoringSetType(coloring,MATCOLORINGSL);CHKERRQ(ierr);
1214     ierr = MatColoringSetFromOptions(coloring);CHKERRQ(ierr);
1215     ierr = MatColoringApply(coloring,&iscoloring);CHKERRQ(ierr);
1216     ierr = MatColoringDestroy(&coloring);CHKERRQ(ierr);
1217     ierr = MatTransposeColoringCreate(C,iscoloring,&matcoloring);CHKERRQ(ierr);
1218 
1219     abt->matcoloring = matcoloring;
1220 
1221     ierr = ISColoringDestroy(&iscoloring);CHKERRQ(ierr);
1222 
1223     /* Create Bt_dense and C_dense = A*Bt_dense */
1224     ierr = MatCreate(PETSC_COMM_SELF,&Bt_dense);CHKERRQ(ierr);
1225     ierr = MatSetSizes(Bt_dense,A->cmap->n,matcoloring->ncolors,A->cmap->n,matcoloring->ncolors);CHKERRQ(ierr);
1226     ierr = MatSetType(Bt_dense,MATSEQDENSE);CHKERRQ(ierr);
1227     ierr = MatSeqDenseSetPreallocation(Bt_dense,NULL);CHKERRQ(ierr);
1228 
1229     Bt_dense->assembled = PETSC_TRUE;
1230     abt->Bt_den         = Bt_dense;
1231 
1232     ierr = MatCreate(PETSC_COMM_SELF,&C_dense);CHKERRQ(ierr);
1233     ierr = MatSetSizes(C_dense,A->rmap->n,matcoloring->ncolors,A->rmap->n,matcoloring->ncolors);CHKERRQ(ierr);
1234     ierr = MatSetType(C_dense,MATSEQDENSE);CHKERRQ(ierr);
1235     ierr = MatSeqDenseSetPreallocation(C_dense,NULL);CHKERRQ(ierr);
1236 
1237     Bt_dense->assembled = PETSC_TRUE;
1238     abt->ABt_den  = C_dense;
1239 
1240 #if defined(PETSC_USE_INFO)
1241     {
1242       Mat_SeqAIJ *c = (Mat_SeqAIJ*)C->data;
1243       ierr = PetscInfo7(C,"Use coloring of C=A*B^T; B^T: %D %D, Bt_dense: %D,%D; Cnz %D / (cm*ncolors %D) = %g\n",B->cmap->n,B->rmap->n,Bt_dense->rmap->n,Bt_dense->cmap->n,c->nz,A->rmap->n*matcoloring->ncolors,(PetscReal)(c->nz)/(A->rmap->n*matcoloring->ncolors));CHKERRQ(ierr);
1244     }
1245 #endif
1246   }
1247   /* clean up */
1248   ierr = MatDestroy(&Bt);CHKERRQ(ierr);
1249   ierr = MatRestoreSymbolicTranspose_SeqAIJ(B,&bti,&btj);CHKERRQ(ierr);
1250   PetscFunctionReturn(0);
1251 }
1252 
1253 PetscErrorCode MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
1254 {
1255   PetscErrorCode      ierr;
1256   Mat_SeqAIJ          *a   =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c=(Mat_SeqAIJ*)C->data;
1257   PetscInt            *ai  =a->i,*aj=a->j,*bi=b->i,*bj=b->j,anzi,bnzj,nexta,nextb,*acol,*bcol,brow;
1258   PetscInt            cm   =C->rmap->n,*ci=c->i,*cj=c->j,i,j,cnzi,*ccol;
1259   PetscLogDouble      flops=0.0;
1260   MatScalar           *aa  =a->a,*aval,*ba=b->a,*bval,*ca,*cval;
1261   Mat_MatMatTransMult *abt;
1262   Mat_Product         *product = C->product;
1263 
1264   PetscFunctionBegin;
1265   if (!product) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Missing product struct");
1266   abt = (Mat_MatMatTransMult *)product->data;
1267   if (!abt) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Missing product struct");
1268   /* clear old values in C */
1269   if (!c->a) {
1270     ierr      = PetscCalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
1271     c->a      = ca;
1272     c->free_a = PETSC_TRUE;
1273   } else {
1274     ca =  c->a;
1275     ierr = PetscArrayzero(ca,ci[cm]+1);CHKERRQ(ierr);
1276   }
1277 
1278   if (abt->usecoloring) {
1279     MatTransposeColoring matcoloring = abt->matcoloring;
1280     Mat                  Bt_dense,C_dense = abt->ABt_den;
1281 
1282     /* Get Bt_dense by Apply MatTransposeColoring to B */
1283     Bt_dense = abt->Bt_den;
1284     ierr = MatTransColoringApplySpToDen(matcoloring,B,Bt_dense);CHKERRQ(ierr);
1285 
1286     /* C_dense = A*Bt_dense */
1287     ierr = MatMatMultNumeric_SeqAIJ_SeqDense(A,Bt_dense,C_dense);CHKERRQ(ierr);
1288 
1289     /* Recover C from C_dense */
1290     ierr = MatTransColoringApplyDenToSp(matcoloring,C_dense,C);CHKERRQ(ierr);
1291     PetscFunctionReturn(0);
1292   }
1293 
1294   for (i=0; i<cm; i++) {
1295     anzi = ai[i+1] - ai[i];
1296     acol = aj + ai[i];
1297     aval = aa + ai[i];
1298     cnzi = ci[i+1] - ci[i];
1299     ccol = cj + ci[i];
1300     cval = ca + ci[i];
1301     for (j=0; j<cnzi; j++) {
1302       brow = ccol[j];
1303       bnzj = bi[brow+1] - bi[brow];
1304       bcol = bj + bi[brow];
1305       bval = ba + bi[brow];
1306 
1307       /* perform sparse inner-product c(i,j)=A[i,:]*B[j,:]^T */
1308       nexta = 0; nextb = 0;
1309       while (nexta<anzi && nextb<bnzj) {
1310         while (nexta < anzi && acol[nexta] < bcol[nextb]) nexta++;
1311         if (nexta == anzi) break;
1312         while (nextb < bnzj && acol[nexta] > bcol[nextb]) nextb++;
1313         if (nextb == bnzj) break;
1314         if (acol[nexta] == bcol[nextb]) {
1315           cval[j] += aval[nexta]*bval[nextb];
1316           nexta++; nextb++;
1317           flops += 2;
1318         }
1319       }
1320     }
1321   }
1322   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1323   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1324   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
1325   PetscFunctionReturn(0);
1326 }
1327 
1328 PetscErrorCode MatDestroy_SeqAIJ_MatTransMatMult(void *data)
1329 {
1330   PetscErrorCode      ierr;
1331   Mat_MatTransMatMult *atb = (Mat_MatTransMatMult*)data;
1332 
1333   PetscFunctionBegin;
1334   ierr = MatDestroy(&atb->At);CHKERRQ(ierr);
1335   if (atb->destroy) {
1336     ierr = (*atb->destroy)(atb->data);CHKERRQ(ierr);
1337   }
1338   ierr = PetscFree(atb);CHKERRQ(ierr);
1339   PetscFunctionReturn(0);
1340 }
1341 
1342 PetscErrorCode MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat C)
1343 {
1344   PetscErrorCode ierr;
1345   Mat            At = NULL;
1346   PetscInt       *ati,*atj;
1347   Mat_Product    *product = C->product;
1348   PetscBool      flg,def,square;
1349 
1350   PetscFunctionBegin;
1351   MatCheckProduct(C,4);
1352   square = (PetscBool)(A == B && A->symmetric && A->symmetric_set);
1353   /* outerproduct */
1354   ierr = PetscStrcmp(product->alg,"outerproduct",&flg);CHKERRQ(ierr);
1355   if (flg) {
1356     /* create symbolic At */
1357     if (!square) {
1358       ierr = MatGetSymbolicTranspose_SeqAIJ(A,&ati,&atj);CHKERRQ(ierr);
1359       ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,A->cmap->n,A->rmap->n,ati,atj,NULL,&At);CHKERRQ(ierr);
1360       ierr = MatSetBlockSizes(At,PetscAbs(A->cmap->bs),PetscAbs(B->cmap->bs));CHKERRQ(ierr);
1361       ierr = MatSetType(At,((PetscObject)A)->type_name);CHKERRQ(ierr);
1362     }
1363     /* get symbolic C=At*B */
1364     ierr = MatProductSetAlgorithm(C,"sorted");CHKERRQ(ierr);
1365     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(square ? A : At,B,fill,C);CHKERRQ(ierr);
1366 
1367     /* clean up */
1368     if (!square) {
1369       ierr = MatDestroy(&At);CHKERRQ(ierr);
1370       ierr = MatRestoreSymbolicTranspose_SeqAIJ(A,&ati,&atj);CHKERRQ(ierr);
1371     }
1372 
1373     C->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ; /* outerproduct */
1374     ierr = MatProductSetAlgorithm(C,"outerproduct");CHKERRQ(ierr);
1375     PetscFunctionReturn(0);
1376   }
1377 
1378   /* matmatmult */
1379   ierr = PetscStrcmp(product->alg,"default",&def);CHKERRQ(ierr);
1380   ierr = PetscStrcmp(product->alg,"at*b",&flg);CHKERRQ(ierr);
1381   if (flg || def) {
1382     Mat_MatTransMatMult *atb;
1383 
1384     if (product->data) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Extra product struct not empty");
1385     ierr = PetscNew(&atb);CHKERRQ(ierr);
1386     if (!square) {
1387       ierr = MatTranspose_SeqAIJ(A,MAT_INITIAL_MATRIX,&At);CHKERRQ(ierr);
1388     }
1389     ierr = MatProductSetAlgorithm(C,"sorted");CHKERRQ(ierr);
1390     ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(square ? A : At,B,fill,C);CHKERRQ(ierr);
1391     ierr = MatProductSetAlgorithm(C,"at*b");CHKERRQ(ierr);
1392     product->data    = atb;
1393     product->destroy = MatDestroy_SeqAIJ_MatTransMatMult;
1394     atb->At          = At;
1395     atb->updateAt    = PETSC_FALSE; /* because At is computed here */
1396 
1397     C->ops->mattransposemultnumeric = NULL; /* see MatProductNumeric_AtB_SeqAIJ_SeqAIJ */
1398     PetscFunctionReturn(0);
1399   }
1400 
1401   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Mat Product Algorithm is not supported");
1402 }
1403 
1404 PetscErrorCode MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
1405 {
1406   PetscErrorCode ierr;
1407   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c=(Mat_SeqAIJ*)C->data;
1408   PetscInt       am=A->rmap->n,anzi,*ai=a->i,*aj=a->j,*bi=b->i,*bj,bnzi,nextb;
1409   PetscInt       cm=C->rmap->n,*ci=c->i,*cj=c->j,crow,*cjj,i,j,k;
1410   PetscLogDouble flops=0.0;
1411   MatScalar      *aa=a->a,*ba,*ca,*caj;
1412 
1413   PetscFunctionBegin;
1414   if (!c->a) {
1415     ierr = PetscCalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
1416 
1417     c->a      = ca;
1418     c->free_a = PETSC_TRUE;
1419   } else {
1420     ca   = c->a;
1421     ierr = PetscArrayzero(ca,ci[cm]);CHKERRQ(ierr);
1422   }
1423 
1424   /* compute A^T*B using outer product (A^T)[:,i]*B[i,:] */
1425   for (i=0; i<am; i++) {
1426     bj   = b->j + bi[i];
1427     ba   = b->a + bi[i];
1428     bnzi = bi[i+1] - bi[i];
1429     anzi = ai[i+1] - ai[i];
1430     for (j=0; j<anzi; j++) {
1431       nextb = 0;
1432       crow  = *aj++;
1433       cjj   = cj + ci[crow];
1434       caj   = ca + ci[crow];
1435       /* perform sparse axpy operation.  Note cjj includes bj. */
1436       for (k=0; nextb<bnzi; k++) {
1437         if (cjj[k] == *(bj+nextb)) { /* ccol == bcol */
1438           caj[k] += (*aa)*(*(ba+nextb));
1439           nextb++;
1440         }
1441       }
1442       flops += 2*bnzi;
1443       aa++;
1444     }
1445   }
1446 
1447   /* Assemble the final matrix and clean up */
1448   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1449   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1450   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
1451   PetscFunctionReturn(0);
1452 }
1453 
1454 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqDense(Mat A,Mat B,PetscReal fill,Mat C)
1455 {
1456   PetscErrorCode ierr;
1457 
1458   PetscFunctionBegin;
1459   ierr = MatMatMultSymbolic_SeqDense_SeqDense(A,B,0.0,C);CHKERRQ(ierr);
1460   C->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqDense;
1461   PetscFunctionReturn(0);
1462 }
1463 
1464 PETSC_INTERN PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat A,Mat B,Mat C,const PetscBool add)
1465 {
1466   Mat_SeqAIJ        *a=(Mat_SeqAIJ*)A->data;
1467   Mat_SeqDense      *bd=(Mat_SeqDense*)B->data;
1468   Mat_SeqDense      *cd=(Mat_SeqDense*)C->data;
1469   PetscErrorCode    ierr;
1470   PetscScalar       *c,r1,r2,r3,r4,*c1,*c2,*c3,*c4;
1471   const PetscScalar *aa,*b,*b1,*b2,*b3,*b4,*av;
1472   const PetscInt    *aj;
1473   PetscInt          cm=C->rmap->n,cn=B->cmap->n,bm=bd->lda,am=A->rmap->n;
1474   PetscInt          clda=cd->lda;
1475   PetscInt          am4=4*clda,bm4=4*bm,col,i,j,n;
1476 
1477   PetscFunctionBegin;
1478   if (!cm || !cn) PetscFunctionReturn(0);
1479   ierr = MatSeqAIJGetArrayRead(A,&av);CHKERRQ(ierr);
1480   if (add) {
1481     ierr = MatDenseGetArray(C,&c);CHKERRQ(ierr);
1482   } else {
1483     ierr = MatDenseGetArrayWrite(C,&c);CHKERRQ(ierr);
1484   }
1485   ierr = MatDenseGetArrayRead(B,&b);CHKERRQ(ierr);
1486   b1 = b; b2 = b1 + bm; b3 = b2 + bm; b4 = b3 + bm;
1487   c1 = c; c2 = c1 + clda; c3 = c2 + clda; c4 = c3 + clda;
1488   for (col=0; col<(cn/4)*4; col += 4) {  /* over columns of C */
1489     for (i=0; i<am; i++) {        /* over rows of A in those columns */
1490       r1 = r2 = r3 = r4 = 0.0;
1491       n  = a->i[i+1] - a->i[i];
1492       aj = a->j + a->i[i];
1493       aa = av + a->i[i];
1494       for (j=0; j<n; j++) {
1495         const PetscScalar aatmp = aa[j];
1496         const PetscInt    ajtmp = aj[j];
1497         r1 += aatmp*b1[ajtmp];
1498         r2 += aatmp*b2[ajtmp];
1499         r3 += aatmp*b3[ajtmp];
1500         r4 += aatmp*b4[ajtmp];
1501       }
1502       if (add) {
1503         c1[i] += r1;
1504         c2[i] += r2;
1505         c3[i] += r3;
1506         c4[i] += r4;
1507       } else {
1508         c1[i] = r1;
1509         c2[i] = r2;
1510         c3[i] = r3;
1511         c4[i] = r4;
1512       }
1513     }
1514     b1 += bm4; b2 += bm4; b3 += bm4; b4 += bm4;
1515     c1 += am4; c2 += am4; c3 += am4; c4 += am4;
1516   }
1517   /* process remaining columns */
1518   if (col != cn) {
1519     PetscInt rc = cn-col;
1520 
1521     if (rc == 1) {
1522       for (i=0; i<am; i++) {
1523         r1 = 0.0;
1524         n  = a->i[i+1] - a->i[i];
1525         aj = a->j + a->i[i];
1526         aa = av + a->i[i];
1527         for (j=0; j<n; j++) r1 += aa[j]*b1[aj[j]];
1528         if (add) c1[i] += r1;
1529         else c1[i] = r1;
1530       }
1531     } else if (rc == 2) {
1532       for (i=0; i<am; i++) {
1533         r1 = r2 = 0.0;
1534         n  = a->i[i+1] - a->i[i];
1535         aj = a->j + a->i[i];
1536         aa = av + a->i[i];
1537         for (j=0; j<n; j++) {
1538           const PetscScalar aatmp = aa[j];
1539           const PetscInt    ajtmp = aj[j];
1540           r1 += aatmp*b1[ajtmp];
1541           r2 += aatmp*b2[ajtmp];
1542         }
1543         if (add) {
1544           c1[i] += r1;
1545           c2[i] += r2;
1546         } else {
1547           c1[i] = r1;
1548           c2[i] = r2;
1549         }
1550       }
1551     } else {
1552       for (i=0; i<am; i++) {
1553         r1 = r2 = r3 = 0.0;
1554         n  = a->i[i+1] - a->i[i];
1555         aj = a->j + a->i[i];
1556         aa = av + a->i[i];
1557         for (j=0; j<n; j++) {
1558           const PetscScalar aatmp = aa[j];
1559           const PetscInt    ajtmp = aj[j];
1560           r1 += aatmp*b1[ajtmp];
1561           r2 += aatmp*b2[ajtmp];
1562           r3 += aatmp*b3[ajtmp];
1563         }
1564         if (add) {
1565           c1[i] += r1;
1566           c2[i] += r2;
1567           c3[i] += r3;
1568         } else {
1569           c1[i] = r1;
1570           c2[i] = r2;
1571           c3[i] = r3;
1572         }
1573       }
1574     }
1575   }
1576   ierr = PetscLogFlops(cn*(2.0*a->nz));CHKERRQ(ierr);
1577   if (add) {
1578     ierr = MatDenseRestoreArray(C,&c);CHKERRQ(ierr);
1579   } else {
1580     ierr = MatDenseRestoreArrayWrite(C,&c);CHKERRQ(ierr);
1581   }
1582   ierr = MatDenseRestoreArrayRead(B,&b);CHKERRQ(ierr);
1583   ierr = MatSeqAIJRestoreArrayRead(A,&av);CHKERRQ(ierr);
1584   PetscFunctionReturn(0);
1585 }
1586 
1587 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqDense(Mat A,Mat B,Mat C)
1588 {
1589   PetscErrorCode ierr;
1590 
1591   PetscFunctionBegin;
1592   if (B->rmap->n != A->cmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Number columns in A %D not equal rows in B %D\n",A->cmap->n,B->rmap->n);
1593   if (A->rmap->n != C->rmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Number rows in C %D not equal rows in A %D\n",C->rmap->n,A->rmap->n);
1594   if (B->cmap->n != C->cmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Number columns in B %D not equal columns in C %D\n",B->cmap->n,C->cmap->n);
1595 
1596   ierr = MatMatMultNumericAdd_SeqAIJ_SeqDense(A,B,C,PETSC_FALSE);CHKERRQ(ierr);
1597   PetscFunctionReturn(0);
1598 }
1599 
1600 /* ------------------------------------------------------- */
1601 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_SeqDense_AB(Mat C)
1602 {
1603   PetscFunctionBegin;
1604   C->ops->matmultsymbolic = MatMatMultSymbolic_SeqAIJ_SeqDense;
1605   C->ops->productsymbolic = MatProductSymbolic_AB;
1606   PetscFunctionReturn(0);
1607 }
1608 
1609 PETSC_INTERN PetscErrorCode MatTMatTMultSymbolic_SeqAIJ_SeqDense(Mat,Mat,PetscReal,Mat);
1610 
1611 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_SeqDense_AtB(Mat C)
1612 {
1613   PetscFunctionBegin;
1614   C->ops->transposematmultsymbolic = MatTMatTMultSymbolic_SeqAIJ_SeqDense;
1615   C->ops->productsymbolic          = MatProductSymbolic_AtB;
1616   PetscFunctionReturn(0);
1617 }
1618 
1619 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_SeqDense_ABt(Mat C)
1620 {
1621   PetscFunctionBegin;
1622   C->ops->mattransposemultsymbolic = MatTMatTMultSymbolic_SeqAIJ_SeqDense;
1623   C->ops->productsymbolic          = MatProductSymbolic_ABt;
1624   PetscFunctionReturn(0);
1625 }
1626 
1627 PETSC_INTERN PetscErrorCode MatProductSetFromOptions_SeqAIJ_SeqDense(Mat C)
1628 {
1629   PetscErrorCode ierr;
1630   Mat_Product    *product = C->product;
1631 
1632   PetscFunctionBegin;
1633   switch (product->type) {
1634   case MATPRODUCT_AB:
1635     ierr = MatProductSetFromOptions_SeqAIJ_SeqDense_AB(C);CHKERRQ(ierr);
1636     break;
1637   case MATPRODUCT_AtB:
1638     ierr = MatProductSetFromOptions_SeqAIJ_SeqDense_AtB(C);CHKERRQ(ierr);
1639     break;
1640   case MATPRODUCT_ABt:
1641     ierr = MatProductSetFromOptions_SeqAIJ_SeqDense_ABt(C);CHKERRQ(ierr);
1642     break;
1643   default:
1644     break;
1645   }
1646   PetscFunctionReturn(0);
1647 }
1648 /* ------------------------------------------------------- */
1649 static PetscErrorCode MatProductSetFromOptions_SeqXBAIJ_SeqDense_AB(Mat C)
1650 {
1651   PetscErrorCode ierr;
1652   Mat_Product    *product = C->product;
1653   Mat            A = product->A;
1654   PetscBool      baij;
1655 
1656   PetscFunctionBegin;
1657   ierr = PetscObjectTypeCompare((PetscObject)A,MATSEQBAIJ,&baij);CHKERRQ(ierr);
1658   if (!baij) { /* A is seqsbaij */
1659     PetscBool sbaij;
1660     ierr = PetscObjectTypeCompare((PetscObject)A,MATSEQSBAIJ,&sbaij);CHKERRQ(ierr);
1661     if (!sbaij) SETERRQ(PetscObjectComm((PetscObject)C),PETSC_ERR_ARG_WRONGSTATE,"Mat must be either seqbaij or seqsbaij format");
1662 
1663     C->ops->matmultsymbolic = MatMatMultSymbolic_SeqSBAIJ_SeqDense;
1664   } else { /* A is seqbaij */
1665     C->ops->matmultsymbolic = MatMatMultSymbolic_SeqBAIJ_SeqDense;
1666   }
1667 
1668   C->ops->productsymbolic = MatProductSymbolic_AB;
1669   PetscFunctionReturn(0);
1670 }
1671 
1672 PETSC_INTERN PetscErrorCode MatProductSetFromOptions_SeqXBAIJ_SeqDense(Mat C)
1673 {
1674   PetscErrorCode ierr;
1675   Mat_Product    *product = C->product;
1676 
1677   PetscFunctionBegin;
1678   MatCheckProduct(C,1);
1679   if (!product->A) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Missing A");
1680   if (product->type == MATPRODUCT_AB || (product->type == MATPRODUCT_AtB && product->A->symmetric)) {
1681     ierr = MatProductSetFromOptions_SeqXBAIJ_SeqDense_AB(C);CHKERRQ(ierr);
1682   }
1683   PetscFunctionReturn(0);
1684 }
1685 
1686 /* ------------------------------------------------------- */
1687 static PetscErrorCode MatProductSetFromOptions_SeqDense_SeqAIJ_AB(Mat C)
1688 {
1689   PetscFunctionBegin;
1690   C->ops->matmultsymbolic = MatMatMultSymbolic_SeqDense_SeqAIJ;
1691   C->ops->productsymbolic = MatProductSymbolic_AB;
1692   PetscFunctionReturn(0);
1693 }
1694 
1695 PETSC_INTERN PetscErrorCode MatProductSetFromOptions_SeqDense_SeqAIJ(Mat C)
1696 {
1697   PetscErrorCode ierr;
1698   Mat_Product    *product = C->product;
1699 
1700   PetscFunctionBegin;
1701   if (product->type == MATPRODUCT_AB) {
1702     ierr = MatProductSetFromOptions_SeqDense_SeqAIJ_AB(C);CHKERRQ(ierr);
1703   }
1704   PetscFunctionReturn(0);
1705 }
1706 /* ------------------------------------------------------- */
1707 
1708 PetscErrorCode  MatTransColoringApplySpToDen_SeqAIJ(MatTransposeColoring coloring,Mat B,Mat Btdense)
1709 {
1710   PetscErrorCode ierr;
1711   Mat_SeqAIJ     *b       = (Mat_SeqAIJ*)B->data;
1712   Mat_SeqDense   *btdense = (Mat_SeqDense*)Btdense->data;
1713   PetscInt       *bi      = b->i,*bj=b->j;
1714   PetscInt       m        = Btdense->rmap->n,n=Btdense->cmap->n,j,k,l,col,anz,*btcol,brow,ncolumns;
1715   MatScalar      *btval,*btval_den,*ba=b->a;
1716   PetscInt       *columns=coloring->columns,*colorforcol=coloring->colorforcol,ncolors=coloring->ncolors;
1717 
1718   PetscFunctionBegin;
1719   btval_den=btdense->v;
1720   ierr     = PetscArrayzero(btval_den,m*n);CHKERRQ(ierr);
1721   for (k=0; k<ncolors; k++) {
1722     ncolumns = coloring->ncolumns[k];
1723     for (l=0; l<ncolumns; l++) { /* insert a row of B to a column of Btdense */
1724       col   = *(columns + colorforcol[k] + l);
1725       btcol = bj + bi[col];
1726       btval = ba + bi[col];
1727       anz   = bi[col+1] - bi[col];
1728       for (j=0; j<anz; j++) {
1729         brow            = btcol[j];
1730         btval_den[brow] = btval[j];
1731       }
1732     }
1733     btval_den += m;
1734   }
1735   PetscFunctionReturn(0);
1736 }
1737 
1738 PetscErrorCode MatTransColoringApplyDenToSp_SeqAIJ(MatTransposeColoring matcoloring,Mat Cden,Mat Csp)
1739 {
1740   PetscErrorCode    ierr;
1741   Mat_SeqAIJ        *csp = (Mat_SeqAIJ*)Csp->data;
1742   const PetscScalar *ca_den,*ca_den_ptr;
1743   PetscScalar       *ca=csp->a;
1744   PetscInt          k,l,m=Cden->rmap->n,ncolors=matcoloring->ncolors;
1745   PetscInt          brows=matcoloring->brows,*den2sp=matcoloring->den2sp;
1746   PetscInt          nrows,*row,*idx;
1747   PetscInt          *rows=matcoloring->rows,*colorforrow=matcoloring->colorforrow;
1748 
1749   PetscFunctionBegin;
1750   ierr   = MatDenseGetArrayRead(Cden,&ca_den);CHKERRQ(ierr);
1751 
1752   if (brows > 0) {
1753     PetscInt *lstart,row_end,row_start;
1754     lstart = matcoloring->lstart;
1755     ierr = PetscArrayzero(lstart,ncolors);CHKERRQ(ierr);
1756 
1757     row_end = brows;
1758     if (row_end > m) row_end = m;
1759     for (row_start=0; row_start<m; row_start+=brows) { /* loop over row blocks of Csp */
1760       ca_den_ptr = ca_den;
1761       for (k=0; k<ncolors; k++) { /* loop over colors (columns of Cden) */
1762         nrows = matcoloring->nrows[k];
1763         row   = rows  + colorforrow[k];
1764         idx   = den2sp + colorforrow[k];
1765         for (l=lstart[k]; l<nrows; l++) {
1766           if (row[l] >= row_end) {
1767             lstart[k] = l;
1768             break;
1769           } else {
1770             ca[idx[l]] = ca_den_ptr[row[l]];
1771           }
1772         }
1773         ca_den_ptr += m;
1774       }
1775       row_end += brows;
1776       if (row_end > m) row_end = m;
1777     }
1778   } else { /* non-blocked impl: loop over columns of Csp - slow if Csp is large */
1779     ca_den_ptr = ca_den;
1780     for (k=0; k<ncolors; k++) {
1781       nrows = matcoloring->nrows[k];
1782       row   = rows  + colorforrow[k];
1783       idx   = den2sp + colorforrow[k];
1784       for (l=0; l<nrows; l++) {
1785         ca[idx[l]] = ca_den_ptr[row[l]];
1786       }
1787       ca_den_ptr += m;
1788     }
1789   }
1790 
1791   ierr = MatDenseRestoreArrayRead(Cden,&ca_den);CHKERRQ(ierr);
1792 #if defined(PETSC_USE_INFO)
1793   if (matcoloring->brows > 0) {
1794     ierr = PetscInfo1(Csp,"Loop over %D row blocks for den2sp\n",brows);CHKERRQ(ierr);
1795   } else {
1796     ierr = PetscInfo(Csp,"Loop over colors/columns of Cden, inefficient for large sparse matrix product \n");CHKERRQ(ierr);
1797   }
1798 #endif
1799   PetscFunctionReturn(0);
1800 }
1801 
1802 PetscErrorCode MatTransposeColoringCreate_SeqAIJ(Mat mat,ISColoring iscoloring,MatTransposeColoring c)
1803 {
1804   PetscErrorCode ierr;
1805   PetscInt       i,n,nrows,Nbs,j,k,m,ncols,col,cm;
1806   const PetscInt *is,*ci,*cj,*row_idx;
1807   PetscInt       nis = iscoloring->n,*rowhit,bs = 1;
1808   IS             *isa;
1809   Mat_SeqAIJ     *csp = (Mat_SeqAIJ*)mat->data;
1810   PetscInt       *colorforrow,*rows,*rows_i,*idxhit,*spidx,*den2sp,*den2sp_i;
1811   PetscInt       *colorforcol,*columns,*columns_i,brows;
1812   PetscBool      flg;
1813 
1814   PetscFunctionBegin;
1815   ierr = ISColoringGetIS(iscoloring,PETSC_USE_POINTER,PETSC_IGNORE,&isa);CHKERRQ(ierr);
1816 
1817   /* bs >1 is not being tested yet! */
1818   Nbs       = mat->cmap->N/bs;
1819   c->M      = mat->rmap->N/bs;  /* set total rows, columns and local rows */
1820   c->N      = Nbs;
1821   c->m      = c->M;
1822   c->rstart = 0;
1823   c->brows  = 100;
1824 
1825   c->ncolors = nis;
1826   ierr = PetscMalloc3(nis,&c->ncolumns,nis,&c->nrows,nis+1,&colorforrow);CHKERRQ(ierr);
1827   ierr = PetscMalloc1(csp->nz+1,&rows);CHKERRQ(ierr);
1828   ierr = PetscMalloc1(csp->nz+1,&den2sp);CHKERRQ(ierr);
1829 
1830   brows = c->brows;
1831   ierr = PetscOptionsGetInt(NULL,NULL,"-matden2sp_brows",&brows,&flg);CHKERRQ(ierr);
1832   if (flg) c->brows = brows;
1833   if (brows > 0) {
1834     ierr = PetscMalloc1(nis+1,&c->lstart);CHKERRQ(ierr);
1835   }
1836 
1837   colorforrow[0] = 0;
1838   rows_i         = rows;
1839   den2sp_i       = den2sp;
1840 
1841   ierr = PetscMalloc1(nis+1,&colorforcol);CHKERRQ(ierr);
1842   ierr = PetscMalloc1(Nbs+1,&columns);CHKERRQ(ierr);
1843 
1844   colorforcol[0] = 0;
1845   columns_i      = columns;
1846 
1847   /* get column-wise storage of mat */
1848   ierr = MatGetColumnIJ_SeqAIJ_Color(mat,0,PETSC_FALSE,PETSC_FALSE,&ncols,&ci,&cj,&spidx,NULL);CHKERRQ(ierr);
1849 
1850   cm   = c->m;
1851   ierr = PetscMalloc1(cm+1,&rowhit);CHKERRQ(ierr);
1852   ierr = PetscMalloc1(cm+1,&idxhit);CHKERRQ(ierr);
1853   for (i=0; i<nis; i++) { /* loop over color */
1854     ierr = ISGetLocalSize(isa[i],&n);CHKERRQ(ierr);
1855     ierr = ISGetIndices(isa[i],&is);CHKERRQ(ierr);
1856 
1857     c->ncolumns[i] = n;
1858     if (n) {
1859       ierr = PetscArraycpy(columns_i,is,n);CHKERRQ(ierr);
1860     }
1861     colorforcol[i+1] = colorforcol[i] + n;
1862     columns_i       += n;
1863 
1864     /* fast, crude version requires O(N*N) work */
1865     ierr = PetscArrayzero(rowhit,cm);CHKERRQ(ierr);
1866 
1867     for (j=0; j<n; j++) { /* loop over columns*/
1868       col     = is[j];
1869       row_idx = cj + ci[col];
1870       m       = ci[col+1] - ci[col];
1871       for (k=0; k<m; k++) { /* loop over columns marking them in rowhit */
1872         idxhit[*row_idx]   = spidx[ci[col] + k];
1873         rowhit[*row_idx++] = col + 1;
1874       }
1875     }
1876     /* count the number of hits */
1877     nrows = 0;
1878     for (j=0; j<cm; j++) {
1879       if (rowhit[j]) nrows++;
1880     }
1881     c->nrows[i]      = nrows;
1882     colorforrow[i+1] = colorforrow[i] + nrows;
1883 
1884     nrows = 0;
1885     for (j=0; j<cm; j++) { /* loop over rows */
1886       if (rowhit[j]) {
1887         rows_i[nrows]   = j;
1888         den2sp_i[nrows] = idxhit[j];
1889         nrows++;
1890       }
1891     }
1892     den2sp_i += nrows;
1893 
1894     ierr    = ISRestoreIndices(isa[i],&is);CHKERRQ(ierr);
1895     rows_i += nrows;
1896   }
1897   ierr = MatRestoreColumnIJ_SeqAIJ_Color(mat,0,PETSC_FALSE,PETSC_FALSE,&ncols,&ci,&cj,&spidx,NULL);CHKERRQ(ierr);
1898   ierr = PetscFree(rowhit);CHKERRQ(ierr);
1899   ierr = ISColoringRestoreIS(iscoloring,PETSC_USE_POINTER,&isa);CHKERRQ(ierr);
1900   if (csp->nz != colorforrow[nis]) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_PLIB,"csp->nz %d != colorforrow[nis] %d",csp->nz,colorforrow[nis]);
1901 
1902   c->colorforrow = colorforrow;
1903   c->rows        = rows;
1904   c->den2sp      = den2sp;
1905   c->colorforcol = colorforcol;
1906   c->columns     = columns;
1907 
1908   ierr = PetscFree(idxhit);CHKERRQ(ierr);
1909   PetscFunctionReturn(0);
1910 }
1911 
1912 /* --------------------------------------------------------------- */
1913 static PetscErrorCode MatProductNumeric_AtB_SeqAIJ_SeqAIJ(Mat C)
1914 {
1915   PetscErrorCode ierr;
1916   Mat_Product    *product = C->product;
1917   Mat            A=product->A,B=product->B;
1918 
1919   PetscFunctionBegin;
1920   if (C->ops->mattransposemultnumeric) {
1921     /* Alg: "outerproduct" */
1922     ierr = (*C->ops->mattransposemultnumeric)(A,B,C);CHKERRQ(ierr);
1923   } else {
1924     /* Alg: "matmatmult" -- C = At*B */
1925     Mat_MatTransMatMult *atb = (Mat_MatTransMatMult *)product->data;
1926     Mat                 At;
1927 
1928     if (!atb) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Missing product struct");
1929     At = atb->At;
1930     if (atb->updateAt && At) { /* At is computed in MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ() */
1931       ierr = MatTranspose_SeqAIJ(A,MAT_REUSE_MATRIX,&At);CHKERRQ(ierr);
1932     }
1933     ierr = MatMatMultNumeric_SeqAIJ_SeqAIJ(At ? At : A,B,C);CHKERRQ(ierr);
1934     atb->updateAt = PETSC_TRUE;
1935   }
1936   PetscFunctionReturn(0);
1937 }
1938 
1939 static PetscErrorCode MatProductSymbolic_AtB_SeqAIJ_SeqAIJ(Mat C)
1940 {
1941   PetscErrorCode ierr;
1942   Mat_Product    *product = C->product;
1943   Mat            A=product->A,B=product->B;
1944   PetscReal      fill=product->fill;
1945 
1946   PetscFunctionBegin;
1947   ierr = MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ(A,B,fill,C);CHKERRQ(ierr);
1948 
1949   C->ops->productnumeric = MatProductNumeric_AtB_SeqAIJ_SeqAIJ;
1950   PetscFunctionReturn(0);
1951 }
1952 
1953 /* --------------------------------------------------------------- */
1954 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_AB(Mat C)
1955 {
1956   PetscErrorCode ierr;
1957   Mat_Product    *product = C->product;
1958   PetscInt       alg = 0; /* default algorithm */
1959   PetscBool      flg = PETSC_FALSE;
1960 #if !defined(PETSC_HAVE_HYPRE)
1961   const char     *algTypes[7] = {"sorted","scalable","scalable_fast","heap","btheap","llcondensed","rowmerge"};
1962   PetscInt       nalg = 7;
1963 #else
1964   const char     *algTypes[8] = {"sorted","scalable","scalable_fast","heap","btheap","llcondensed","rowmerge","hypre"};
1965   PetscInt       nalg = 8;
1966 #endif
1967 
1968   PetscFunctionBegin;
1969   /* Set default algorithm */
1970   ierr = PetscStrcmp(C->product->alg,"default",&flg);CHKERRQ(ierr);
1971   if (flg) {
1972     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
1973   }
1974 
1975   /* Get runtime option */
1976   if (product->api_user) {
1977     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatMatMult","Mat");CHKERRQ(ierr);
1978     ierr = PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[0],&alg,&flg);CHKERRQ(ierr);
1979     ierr = PetscOptionsEnd();CHKERRQ(ierr);
1980   } else {
1981     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_AB","Mat");CHKERRQ(ierr);
1982     ierr = PetscOptionsEList("-matproduct_ab_via","Algorithmic approach","MatProduct_AB",algTypes,nalg,algTypes[0],&alg,&flg);CHKERRQ(ierr);
1983     ierr = PetscOptionsEnd();CHKERRQ(ierr);
1984   }
1985   if (flg) {
1986     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
1987   }
1988 
1989   C->ops->productsymbolic = MatProductSymbolic_AB;
1990   C->ops->matmultsymbolic = MatMatMultSymbolic_SeqAIJ_SeqAIJ;
1991   PetscFunctionReturn(0);
1992 }
1993 
1994 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_AtB(Mat C)
1995 {
1996   PetscErrorCode ierr;
1997   Mat_Product    *product = C->product;
1998   PetscInt       alg = 0; /* default algorithm */
1999   PetscBool      flg = PETSC_FALSE;
2000   const char     *algTypes[3] = {"default","at*b","outerproduct"};
2001   PetscInt       nalg = 3;
2002 
2003   PetscFunctionBegin;
2004   /* Get runtime option */
2005   if (product->api_user) {
2006     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatTransposeMatMult","Mat");CHKERRQ(ierr);
2007     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2008     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2009   } else {
2010     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_AtB","Mat");CHKERRQ(ierr);
2011     ierr = PetscOptionsEList("-matproduct_atb_via","Algorithmic approach","MatProduct_AtB",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2012     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2013   }
2014   if (flg) {
2015     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2016   }
2017 
2018   C->ops->productsymbolic = MatProductSymbolic_AtB_SeqAIJ_SeqAIJ;
2019   PetscFunctionReturn(0);
2020 }
2021 
2022 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_ABt(Mat C)
2023 {
2024   PetscErrorCode ierr;
2025   Mat_Product    *product = C->product;
2026   PetscInt       alg = 0; /* default algorithm */
2027   PetscBool      flg = PETSC_FALSE;
2028   const char     *algTypes[2] = {"default","color"};
2029   PetscInt       nalg = 2;
2030 
2031   PetscFunctionBegin;
2032   /* Set default algorithm */
2033   ierr = PetscStrcmp(C->product->alg,"default",&flg);CHKERRQ(ierr);
2034   if (!flg) {
2035     alg = 1;
2036     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2037   }
2038 
2039   /* Get runtime option */
2040   if (product->api_user) {
2041     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatMatTransposeMult","Mat");CHKERRQ(ierr);
2042     ierr = PetscOptionsEList("-matmattransmult_via","Algorithmic approach","MatMatTransposeMult",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2043     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2044   } else {
2045     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_ABt","Mat");CHKERRQ(ierr);
2046     ierr = PetscOptionsEList("-matproduct_abt_via","Algorithmic approach","MatProduct_ABt",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2047     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2048   }
2049   if (flg) {
2050     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2051   }
2052 
2053   C->ops->mattransposemultsymbolic = MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ;
2054   C->ops->productsymbolic          = MatProductSymbolic_ABt;
2055   PetscFunctionReturn(0);
2056 }
2057 
2058 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_PtAP(Mat C)
2059 {
2060   PetscErrorCode ierr;
2061   Mat_Product    *product = C->product;
2062   PetscBool      flg = PETSC_FALSE;
2063   PetscInt       alg = 0; /* default algorithm -- alg=1 should be default!!! */
2064 #if !defined(PETSC_HAVE_HYPRE)
2065   const char      *algTypes[2] = {"scalable","rap"};
2066   PetscInt        nalg = 2;
2067 #else
2068   const char      *algTypes[3] = {"scalable","rap","hypre"};
2069   PetscInt        nalg = 3;
2070 #endif
2071 
2072   PetscFunctionBegin;
2073   /* Set default algorithm */
2074   ierr = PetscStrcmp(product->alg,"default",&flg);CHKERRQ(ierr);
2075   if (flg) {
2076     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2077   }
2078 
2079   /* Get runtime option */
2080   if (product->api_user) {
2081     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatPtAP","Mat");CHKERRQ(ierr);
2082     ierr = PetscOptionsEList("-matptap_via","Algorithmic approach","MatPtAP",algTypes,nalg,algTypes[0],&alg,&flg);CHKERRQ(ierr);
2083     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2084   } else {
2085     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_PtAP","Mat");CHKERRQ(ierr);
2086     ierr = PetscOptionsEList("-matproduct_ptap_via","Algorithmic approach","MatProduct_PtAP",algTypes,nalg,algTypes[0],&alg,&flg);CHKERRQ(ierr);
2087     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2088   }
2089   if (flg) {
2090     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2091   }
2092 
2093   C->ops->productsymbolic = MatProductSymbolic_PtAP_SeqAIJ_SeqAIJ;
2094   PetscFunctionReturn(0);
2095 }
2096 
2097 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_RARt(Mat C)
2098 {
2099   PetscErrorCode ierr;
2100   Mat_Product    *product = C->product;
2101   PetscBool      flg = PETSC_FALSE;
2102   PetscInt       alg = 0; /* default algorithm */
2103   const char     *algTypes[3] = {"r*a*rt","r*art","coloring_rart"};
2104   PetscInt        nalg = 3;
2105 
2106   PetscFunctionBegin;
2107   /* Set default algorithm */
2108   ierr = PetscStrcmp(product->alg,"default",&flg);CHKERRQ(ierr);
2109   if (flg) {
2110     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2111   }
2112 
2113   /* Get runtime option */
2114   if (product->api_user) {
2115     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatRARt","Mat");CHKERRQ(ierr);
2116     ierr = PetscOptionsEList("-matrart_via","Algorithmic approach","MatRARt",algTypes,nalg,algTypes[0],&alg,&flg);CHKERRQ(ierr);
2117     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2118   } else {
2119     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_RARt","Mat");CHKERRQ(ierr);
2120     ierr = PetscOptionsEList("-matproduct_rart_via","Algorithmic approach","MatProduct_RARt",algTypes,nalg,algTypes[0],&alg,&flg);CHKERRQ(ierr);
2121     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2122   }
2123   if (flg) {
2124     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2125   }
2126 
2127   C->ops->productsymbolic = MatProductSymbolic_RARt_SeqAIJ_SeqAIJ;
2128   PetscFunctionReturn(0);
2129 }
2130 
2131 /* ABC = A*B*C = A*(B*C); ABC's algorithm must be chosen from AB's algorithm */
2132 static PetscErrorCode MatProductSetFromOptions_SeqAIJ_ABC(Mat C)
2133 {
2134   PetscErrorCode ierr;
2135   Mat_Product    *product = C->product;
2136   PetscInt       alg = 0; /* default algorithm */
2137   PetscBool      flg = PETSC_FALSE;
2138   const char     *algTypes[7] = {"sorted","scalable","scalable_fast","heap","btheap","llcondensed","rowmerge"};
2139   PetscInt       nalg = 7;
2140 
2141   PetscFunctionBegin;
2142   /* Set default algorithm */
2143   ierr = PetscStrcmp(product->alg,"default",&flg);CHKERRQ(ierr);
2144   if (flg) {
2145     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2146   }
2147 
2148   /* Get runtime option */
2149   if (product->api_user) {
2150     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatMatMatMult","Mat");CHKERRQ(ierr);
2151     ierr = PetscOptionsEList("-matmatmatmult_via","Algorithmic approach","MatMatMatMult",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2152     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2153   } else {
2154     ierr = PetscOptionsBegin(PetscObjectComm((PetscObject)C),((PetscObject)C)->prefix,"MatProduct_ABC","Mat");CHKERRQ(ierr);
2155     ierr = PetscOptionsEList("-matproduct_abc_via","Algorithmic approach","MatProduct_ABC",algTypes,nalg,algTypes[alg],&alg,&flg);CHKERRQ(ierr);
2156     ierr = PetscOptionsEnd();CHKERRQ(ierr);
2157   }
2158   if (flg) {
2159     ierr = MatProductSetAlgorithm(C,(MatProductAlgorithm)algTypes[alg]);CHKERRQ(ierr);
2160   }
2161 
2162   C->ops->matmatmultsymbolic = MatMatMatMultSymbolic_SeqAIJ_SeqAIJ_SeqAIJ;
2163   C->ops->productsymbolic    = MatProductSymbolic_ABC;
2164   PetscFunctionReturn(0);
2165 }
2166 
2167 PetscErrorCode MatProductSetFromOptions_SeqAIJ(Mat C)
2168 {
2169   PetscErrorCode ierr;
2170   Mat_Product    *product = C->product;
2171 
2172   PetscFunctionBegin;
2173   switch (product->type) {
2174   case MATPRODUCT_AB:
2175     ierr = MatProductSetFromOptions_SeqAIJ_AB(C);CHKERRQ(ierr);
2176     break;
2177   case MATPRODUCT_AtB:
2178     ierr = MatProductSetFromOptions_SeqAIJ_AtB(C);CHKERRQ(ierr);
2179     break;
2180   case MATPRODUCT_ABt:
2181     ierr = MatProductSetFromOptions_SeqAIJ_ABt(C);CHKERRQ(ierr);
2182     break;
2183   case MATPRODUCT_PtAP:
2184     ierr = MatProductSetFromOptions_SeqAIJ_PtAP(C);CHKERRQ(ierr);
2185     break;
2186   case MATPRODUCT_RARt:
2187     ierr = MatProductSetFromOptions_SeqAIJ_RARt(C);CHKERRQ(ierr);
2188     break;
2189   case MATPRODUCT_ABC:
2190     ierr = MatProductSetFromOptions_SeqAIJ_ABC(C);CHKERRQ(ierr);
2191     break;
2192   default:
2193     break;
2194   }
2195   PetscFunctionReturn(0);
2196 }
2197