xref: /petsc/src/mat/impls/aij/seq/matmatmult.c (revision e1e80dc898c3d5fa028e909e947004e745fb92d9)
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  static PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(Mat,Mat,PetscReal,Mat*);
14 
15  #if defined(PETSC_HAVE_HYPRE)
16  PETSC_INTERN PetscErrorCode MatMatMultSymbolic_AIJ_AIJ_wHYPRE(Mat,Mat,PetscReal,Mat*);
17  #endif
18 
19  PETSC_INTERN PetscErrorCode MatMatMult_SeqAIJ_SeqAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
20  {
21    PetscErrorCode ierr;
22  #if !defined(PETSC_HAVE_HYPRE)
23    const char     *algTypes[7] = {"sorted","scalable","scalable_fast","heap","btheap","llcondensed","combined"};
24    PetscInt       nalg = 7;
25  #else
26    const char     *algTypes[8] = {"sorted","scalable","scalable_fast","heap","btheap","llcondensed","combined","hypre"};
27    PetscInt       nalg = 8;
28  #endif
29    PetscInt       alg = 0; /* set default algorithm */
30    PetscBool      combined = PETSC_FALSE;  /* Indicates whether the symbolic stage already computed the numerical values. */
31 
32    PetscFunctionBegin;
33    if (scall == MAT_INITIAL_MATRIX) {
34      ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
35      PetscOptionsObject->alreadyprinted = PETSC_FALSE; /* a hack to ensure the option shows in '-help' */
36      ierr = PetscOptionsEList("-matmatmult_via","Algorithmic approach","MatMatMult",algTypes,nalg,algTypes[0],&alg,NULL);CHKERRQ(ierr);
37      ierr = PetscOptionsEnd();CHKERRQ(ierr);
38      ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
39      switch (alg) {
40      case 1:
41        ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable(A,B,fill,C);CHKERRQ(ierr);
42        break;
43      case 2:
44        ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable_fast(A,B,fill,C);CHKERRQ(ierr);
45        break;
46      case 3:
47        ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Heap(A,B,fill,C);CHKERRQ(ierr);
48        break;
49      case 4:
50        ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_BTHeap(A,B,fill,C);CHKERRQ(ierr);
51        break;
52      case 5:
53        ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(A,B,fill,C);CHKERRQ(ierr);
54        break;
55      case 6:
56        ierr = MatMatMult_SeqAIJ_SeqAIJ_Combined(A,B,fill,C);CHKERRQ(ierr);
57        combined = PETSC_TRUE;
58        break;
59  #if defined(PETSC_HAVE_HYPRE)
60      case 7:
61        ierr = MatMatMultSymbolic_AIJ_AIJ_wHYPRE(A,B,fill,C);CHKERRQ(ierr);
62        break;
63  #endif
64      default:
65        ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(A,B,fill,C);CHKERRQ(ierr);
66       break;
67      }
68      ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
69    }
70 
71    ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
72    if (!combined) {
73      ierr = (*(*C)->ops->matmultnumeric)(A,B,*C);CHKERRQ(ierr);
74    }
75    ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
76    PetscFunctionReturn(0);
77  }
78 
79  static PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(Mat A,Mat B,PetscReal fill,Mat *C)
80  {
81    PetscErrorCode     ierr;
82    Mat_SeqAIJ         *a =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
83    PetscInt           *ai=a->i,*bi=b->i,*ci,*cj;
84    PetscInt           am =A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
85    PetscReal          afill;
86    PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,*lnk,ndouble=0,Crmax;
87    PetscTable         ta;
88    PetscBT            lnkbt;
89    PetscFreeSpaceList free_space=NULL,current_space=NULL;
90 
91    PetscFunctionBegin;
92    /* Get ci and cj */
93    /*---------------*/
94    /* Allocate ci array, arrays for fill computation and */
95    /* free space for accumulating nonzero column info */
96    ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
97    ci[0] = 0;
98 
99    /* create and initialize a linked list */
100    ierr = PetscTableCreate(bn,bn,&ta);CHKERRQ(ierr);
101    MatRowMergeMax_SeqAIJ(b,bm,ta);
102    ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
103    ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
104 
105    ierr = PetscLLCondensedCreate(Crmax,bn,&lnk,&lnkbt);CHKERRQ(ierr);
106 
107    /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
108    ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
109 
110    current_space = free_space;
111 
112    /* Determine ci and cj */
113    for (i=0; i<am; i++) {
114      anzi = ai[i+1] - ai[i];
115      aj   = a->j + ai[i];
116      for (j=0; j<anzi; j++) {
117        brow = aj[j];
118        bnzj = bi[brow+1] - bi[brow];
119        bj   = b->j + bi[brow];
120        /* add non-zero cols of B into the sorted linked list lnk */
121        ierr = PetscLLCondensedAddSorted(bnzj,bj,lnk,lnkbt);CHKERRQ(ierr);
122      }
123      cnzi = lnk[0];
124 
125      /* If free space is not available, make more free space */
126      /* Double the amount of total space in the list */
127      if (current_space->local_remaining<cnzi) {
128        ierr = PetscFreeSpaceGet(PetscIntSumTruncate(cnzi,current_space->total_array_size),&current_space);CHKERRQ(ierr);
129        ndouble++;
130      }
131 
132      /* Copy data into free space, then initialize lnk */
133      ierr = PetscLLCondensedClean(bn,cnzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
134 
135      current_space->array           += cnzi;
136      current_space->local_used      += cnzi;
137      current_space->local_remaining -= cnzi;
138 
139      ci[i+1] = ci[i] + cnzi;
140    }
141 
142    /* Column indices are in the list of free space */
143    /* Allocate space for cj, initialize cj, and */
144    /* destroy list of free space and other temporary array(s) */
145    ierr = PetscMalloc1(ci[am]+1,&cj);CHKERRQ(ierr);
146    ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
147    ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
148 
149    /* put together the new symbolic matrix */
150    ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
151    ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
152    ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
153 
154   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
155   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
156   c                         = (Mat_SeqAIJ*)((*C)->data);
157   c->free_a                 = PETSC_FALSE;
158   c->free_ij                = PETSC_TRUE;
159   c->nonew                  = 0;
160   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ; /* fast, needs non-scalable O(bn) array 'abdense' */
161 
162   /* set MatInfo */
163   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
164   if (afill < 1.0) afill = 1.0;
165   c->maxnz                     = ci[am];
166   c->nz                        = ci[am];
167   (*C)->info.mallocs           = ndouble;
168   (*C)->info.fill_ratio_given  = fill;
169   (*C)->info.fill_ratio_needed = afill;
170 
171 #if defined(PETSC_USE_INFO)
172   if (ci[am]) {
173     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
174     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
175   } else {
176     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
177   }
178 #endif
179   PetscFunctionReturn(0);
180 }
181 
182 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
183 {
184   PetscErrorCode ierr;
185   PetscLogDouble flops=0.0;
186   Mat_SeqAIJ     *a   = (Mat_SeqAIJ*)A->data;
187   Mat_SeqAIJ     *b   = (Mat_SeqAIJ*)B->data;
188   Mat_SeqAIJ     *c   = (Mat_SeqAIJ*)C->data;
189   PetscInt       *ai  =a->i,*aj=a->j,*bi=b->i,*bj=b->j,*bjj,*ci=c->i,*cj=c->j;
190   PetscInt       am   =A->rmap->n,cm=C->rmap->n;
191   PetscInt       i,j,k,anzi,bnzi,cnzi,brow;
192   PetscScalar    *aa=a->a,*ba=b->a,*baj,*ca,valtmp;
193   PetscScalar    *ab_dense;
194 
195   PetscFunctionBegin;
196   if (!c->a) { /* first call of MatMatMultNumeric_SeqAIJ_SeqAIJ, allocate ca and matmult_abdense */
197     ierr      = PetscMalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
198     c->a      = ca;
199     c->free_a = PETSC_TRUE;
200   } else {
201     ca        = c->a;
202   }
203   if (!c->matmult_abdense) {
204     ierr = PetscCalloc1(B->cmap->N,&ab_dense);CHKERRQ(ierr);
205     c->matmult_abdense = ab_dense;
206   } else {
207     ab_dense = c->matmult_abdense;
208   }
209 
210   /* clean old values in C */
211   ierr = PetscMemzero(ca,ci[cm]*sizeof(MatScalar));CHKERRQ(ierr);
212   /* Traverse A row-wise. */
213   /* Build the ith row in C by summing over nonzero columns in A, */
214   /* the rows of B corresponding to nonzeros of A. */
215   for (i=0; i<am; i++) {
216     anzi = ai[i+1] - ai[i];
217     for (j=0; j<anzi; j++) {
218       brow = aj[j];
219       bnzi = bi[brow+1] - bi[brow];
220       bjj  = bj + bi[brow];
221       baj  = ba + bi[brow];
222       /* perform dense axpy */
223       valtmp = aa[j];
224       for (k=0; k<bnzi; k++) {
225         ab_dense[bjj[k]] += valtmp*baj[k];
226       }
227       flops += 2*bnzi;
228     }
229     aj += anzi; aa += anzi;
230 
231     cnzi = ci[i+1] - ci[i];
232     for (k=0; k<cnzi; k++) {
233       ca[k]          += ab_dense[cj[k]];
234       ab_dense[cj[k]] = 0.0; /* zero ab_dense */
235     }
236     flops += cnzi;
237     cj    += cnzi; ca += cnzi;
238   }
239   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
240   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
241   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
242   PetscFunctionReturn(0);
243 }
244 
245 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable(Mat A,Mat B,Mat C)
246 {
247   PetscErrorCode ierr;
248   PetscLogDouble flops=0.0;
249   Mat_SeqAIJ     *a   = (Mat_SeqAIJ*)A->data;
250   Mat_SeqAIJ     *b   = (Mat_SeqAIJ*)B->data;
251   Mat_SeqAIJ     *c   = (Mat_SeqAIJ*)C->data;
252   PetscInt       *ai  = a->i,*aj=a->j,*bi=b->i,*bj=b->j,*bjj,*ci=c->i,*cj=c->j;
253   PetscInt       am   = A->rmap->N,cm=C->rmap->N;
254   PetscInt       i,j,k,anzi,bnzi,cnzi,brow;
255   PetscScalar    *aa=a->a,*ba=b->a,*baj,*ca=c->a,valtmp;
256   PetscInt       nextb;
257 
258   PetscFunctionBegin;
259   if (!ca) { /* first call of MatMatMultNumeric_SeqAIJ_SeqAIJ, allocate ca and matmult_abdense */
260     ierr      = PetscMalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
261     c->a      = ca;
262     c->free_a = PETSC_TRUE;
263   }
264 
265   /* clean old values in C */
266   ierr = PetscMemzero(ca,ci[cm]*sizeof(MatScalar));CHKERRQ(ierr);
267   /* Traverse A row-wise. */
268   /* Build the ith row in C by summing over nonzero columns in A, */
269   /* the rows of B corresponding to nonzeros of A. */
270   for (i=0; i<am; i++) {
271     anzi = ai[i+1] - ai[i];
272     cnzi = ci[i+1] - ci[i];
273     for (j=0; j<anzi; j++) {
274       brow = aj[j];
275       bnzi = bi[brow+1] - bi[brow];
276       bjj  = bj + bi[brow];
277       baj  = ba + bi[brow];
278       /* perform sparse axpy */
279       valtmp = aa[j];
280       nextb  = 0;
281       for (k=0; nextb<bnzi; k++) {
282         if (cj[k] == bjj[nextb]) { /* ccol == bcol */
283           ca[k] += valtmp*baj[nextb++];
284         }
285       }
286       flops += 2*bnzi;
287     }
288     aj += anzi; aa += anzi;
289     cj += cnzi; ca += cnzi;
290   }
291 
292   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
293   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
294   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
295   PetscFunctionReturn(0);
296 }
297 
298 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable_fast(Mat A,Mat B,PetscReal fill,Mat *C)
299 {
300   PetscErrorCode     ierr;
301   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
302   PetscInt           *ai = a->i,*bi=b->i,*ci,*cj;
303   PetscInt           am  = A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
304   MatScalar          *ca;
305   PetscReal          afill;
306   PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,*lnk,ndouble=0,Crmax;
307   PetscTable         ta;
308   PetscFreeSpaceList free_space=NULL,current_space=NULL;
309 
310   PetscFunctionBegin;
311   /* Get ci and cj - same as MatMatMultSymbolic_SeqAIJ_SeqAIJ except using PetscLLxxx_fast() */
312   /*-----------------------------------------------------------------------------------------*/
313   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
314   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
315   ci[0] = 0;
316 
317   /* create and initialize a linked list */
318   ierr = PetscTableCreate(bn,bn,&ta);CHKERRQ(ierr);
319   MatRowMergeMax_SeqAIJ(b,bm,ta);
320   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
321   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
322 
323   ierr = PetscLLCondensedCreate_fast(Crmax,&lnk);CHKERRQ(ierr);
324 
325   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
326   ierr          = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
327   current_space = free_space;
328 
329   /* Determine ci and cj */
330   for (i=0; i<am; i++) {
331     anzi = ai[i+1] - ai[i];
332     aj   = a->j + ai[i];
333     for (j=0; j<anzi; j++) {
334       brow = aj[j];
335       bnzj = bi[brow+1] - bi[brow];
336       bj   = b->j + bi[brow];
337       /* add non-zero cols of B into the sorted linked list lnk */
338       ierr = PetscLLCondensedAddSorted_fast(bnzj,bj,lnk);CHKERRQ(ierr);
339     }
340     cnzi = lnk[1];
341 
342     /* If free space is not available, make more free space */
343     /* Double the amount of total space in the list */
344     if (current_space->local_remaining<cnzi) {
345       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(cnzi,current_space->total_array_size),&current_space);CHKERRQ(ierr);
346       ndouble++;
347     }
348 
349     /* Copy data into free space, then initialize lnk */
350     ierr = PetscLLCondensedClean_fast(cnzi,current_space->array,lnk);CHKERRQ(ierr);
351 
352     current_space->array           += cnzi;
353     current_space->local_used      += cnzi;
354     current_space->local_remaining -= cnzi;
355 
356     ci[i+1] = ci[i] + cnzi;
357   }
358 
359   /* Column indices are in the list of free space */
360   /* Allocate space for cj, initialize cj, and */
361   /* destroy list of free space and other temporary array(s) */
362   ierr = PetscMalloc1(ci[am]+1,&cj);CHKERRQ(ierr);
363   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
364   ierr = PetscLLCondensedDestroy_fast(lnk);CHKERRQ(ierr);
365 
366   /* Allocate space for ca */
367   ierr = PetscMalloc1(ci[am]+1,&ca);CHKERRQ(ierr);
368   ierr = PetscMemzero(ca,(ci[am]+1)*sizeof(MatScalar));CHKERRQ(ierr);
369 
370   /* put together the new symbolic matrix */
371   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,ca,C);CHKERRQ(ierr);
372   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
373   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
374 
375   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
376   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
377   c          = (Mat_SeqAIJ*)((*C)->data);
378   c->free_a  = PETSC_TRUE;
379   c->free_ij = PETSC_TRUE;
380   c->nonew   = 0;
381 
382   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable; /* slower, less memory */
383 
384   /* set MatInfo */
385   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
386   if (afill < 1.0) afill = 1.0;
387   c->maxnz                     = ci[am];
388   c->nz                        = ci[am];
389   (*C)->info.mallocs           = ndouble;
390   (*C)->info.fill_ratio_given  = fill;
391   (*C)->info.fill_ratio_needed = afill;
392 
393 #if defined(PETSC_USE_INFO)
394   if (ci[am]) {
395     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
396     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
397   } else {
398     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
399   }
400 #endif
401   PetscFunctionReturn(0);
402 }
403 
404 
405 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable(Mat A,Mat B,PetscReal fill,Mat *C)
406 {
407   PetscErrorCode     ierr;
408   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
409   PetscInt           *ai = a->i,*bi=b->i,*ci,*cj;
410   PetscInt           am  = A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
411   MatScalar          *ca;
412   PetscReal          afill;
413   PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,*lnk,ndouble=0,Crmax;
414   PetscTable         ta;
415   PetscFreeSpaceList free_space=NULL,current_space=NULL;
416 
417   PetscFunctionBegin;
418   /* Get ci and cj - same as MatMatMultSymbolic_SeqAIJ_SeqAIJ except using PetscLLxxx_Scalalbe() */
419   /*---------------------------------------------------------------------------------------------*/
420   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
421   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
422   ci[0] = 0;
423 
424   /* create and initialize a linked list */
425   ierr = PetscTableCreate(bn,bn,&ta);CHKERRQ(ierr);
426   MatRowMergeMax_SeqAIJ(b,bm,ta);
427   ierr = PetscTableGetCount(ta,&Crmax);CHKERRQ(ierr);
428   ierr = PetscTableDestroy(&ta);CHKERRQ(ierr);
429   ierr = PetscLLCondensedCreate_Scalable(Crmax,&lnk);CHKERRQ(ierr);
430 
431   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
432   ierr          = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
433   current_space = free_space;
434 
435   /* Determine ci and cj */
436   for (i=0; i<am; i++) {
437     anzi = ai[i+1] - ai[i];
438     aj   = a->j + ai[i];
439     for (j=0; j<anzi; j++) {
440       brow = aj[j];
441       bnzj = bi[brow+1] - bi[brow];
442       bj   = b->j + bi[brow];
443       /* add non-zero cols of B into the sorted linked list lnk */
444       ierr = PetscLLCondensedAddSorted_Scalable(bnzj,bj,lnk);CHKERRQ(ierr);
445     }
446     cnzi = lnk[0];
447 
448     /* If free space is not available, make more free space */
449     /* Double the amount of total space in the list */
450     if (current_space->local_remaining<cnzi) {
451       ierr = PetscFreeSpaceGet(PetscIntSumTruncate(cnzi,current_space->total_array_size),&current_space);CHKERRQ(ierr);
452       ndouble++;
453     }
454 
455     /* Copy data into free space, then initialize lnk */
456     ierr = PetscLLCondensedClean_Scalable(cnzi,current_space->array,lnk);CHKERRQ(ierr);
457 
458     current_space->array           += cnzi;
459     current_space->local_used      += cnzi;
460     current_space->local_remaining -= cnzi;
461 
462     ci[i+1] = ci[i] + cnzi;
463   }
464 
465   /* Column indices are in the list of free space */
466   /* Allocate space for cj, initialize cj, and */
467   /* destroy list of free space and other temporary array(s) */
468   ierr = PetscMalloc1(ci[am]+1,&cj);CHKERRQ(ierr);
469   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
470   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
471 
472   /* Allocate space for ca */
473   /*-----------------------*/
474   ierr = PetscMalloc1(ci[am]+1,&ca);CHKERRQ(ierr);
475   ierr = PetscMemzero(ca,(ci[am]+1)*sizeof(MatScalar));CHKERRQ(ierr);
476 
477   /* put together the new symbolic matrix */
478   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,ca,C);CHKERRQ(ierr);
479   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
480   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
481 
482   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
483   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
484   c          = (Mat_SeqAIJ*)((*C)->data);
485   c->free_a  = PETSC_TRUE;
486   c->free_ij = PETSC_TRUE;
487   c->nonew   = 0;
488 
489   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable; /* slower, less memory */
490 
491   /* set MatInfo */
492   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
493   if (afill < 1.0) afill = 1.0;
494   c->maxnz                     = ci[am];
495   c->nz                        = ci[am];
496   (*C)->info.mallocs           = ndouble;
497   (*C)->info.fill_ratio_given  = fill;
498   (*C)->info.fill_ratio_needed = afill;
499 
500 #if defined(PETSC_USE_INFO)
501   if (ci[am]) {
502     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
503     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
504   } else {
505     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
506   }
507 #endif
508   PetscFunctionReturn(0);
509 }
510 
511 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Heap(Mat A,Mat B,PetscReal fill,Mat *C)
512 {
513   PetscErrorCode     ierr;
514   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
515   const PetscInt     *ai=a->i,*bi=b->i,*aj=a->j,*bj=b->j;
516   PetscInt           *ci,*cj,*bb;
517   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
518   PetscReal          afill;
519   PetscInt           i,j,col,ndouble = 0;
520   PetscFreeSpaceList free_space=NULL,current_space=NULL;
521   PetscHeap          h;
522 
523   PetscFunctionBegin;
524   /* Get ci and cj - by merging sorted rows using a heap */
525   /*---------------------------------------------------------------------------------------------*/
526   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
527   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
528   ci[0] = 0;
529 
530   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
531   ierr          = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
532   current_space = free_space;
533 
534   ierr = PetscHeapCreate(a->rmax,&h);CHKERRQ(ierr);
535   ierr = PetscMalloc1(a->rmax,&bb);CHKERRQ(ierr);
536 
537   /* Determine ci and cj */
538   for (i=0; i<am; i++) {
539     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 */
540     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
541     ci[i+1] = ci[i];
542     /* Populate the min heap */
543     for (j=0; j<anzi; j++) {
544       bb[j] = bi[acol[j]];         /* bb points at the start of the row */
545       if (bb[j] < bi[acol[j]+1]) { /* Add if row is nonempty */
546         ierr = PetscHeapAdd(h,j,bj[bb[j]++]);CHKERRQ(ierr);
547       }
548     }
549     /* Pick off the min element, adding it to free space */
550     ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
551     while (j >= 0) {
552       if (current_space->local_remaining < 1) { /* double the size, but don't exceed 16 MiB */
553         ierr = PetscFreeSpaceGet(PetscMin(PetscIntMultTruncate(2,current_space->total_array_size),16 << 20),&current_space);CHKERRQ(ierr);
554         ndouble++;
555       }
556       *(current_space->array++) = col;
557       current_space->local_used++;
558       current_space->local_remaining--;
559       ci[i+1]++;
560 
561       /* stash if anything else remains in this row of B */
562       if (bb[j] < bi[acol[j]+1]) {ierr = PetscHeapStash(h,j,bj[bb[j]++]);CHKERRQ(ierr);}
563       while (1) {               /* pop and stash any other rows of B that also had an entry in this column */
564         PetscInt j2,col2;
565         ierr = PetscHeapPeek(h,&j2,&col2);CHKERRQ(ierr);
566         if (col2 != col) break;
567         ierr = PetscHeapPop(h,&j2,&col2);CHKERRQ(ierr);
568         if (bb[j2] < bi[acol[j2]+1]) {ierr = PetscHeapStash(h,j2,bj[bb[j2]++]);CHKERRQ(ierr);}
569       }
570       /* Put any stashed elements back into the min heap */
571       ierr = PetscHeapUnstash(h);CHKERRQ(ierr);
572       ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
573     }
574   }
575   ierr = PetscFree(bb);CHKERRQ(ierr);
576   ierr = PetscHeapDestroy(&h);CHKERRQ(ierr);
577 
578   /* Column indices are in the list of free space */
579   /* Allocate space for cj, initialize cj, and */
580   /* destroy list of free space and other temporary array(s) */
581   ierr = PetscMalloc1(ci[am],&cj);CHKERRQ(ierr);
582   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
583 
584   /* put together the new symbolic matrix */
585   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
586   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
587   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
588 
589   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
590   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
591   c          = (Mat_SeqAIJ*)((*C)->data);
592   c->free_a  = PETSC_TRUE;
593   c->free_ij = PETSC_TRUE;
594   c->nonew   = 0;
595 
596   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ;
597 
598   /* set MatInfo */
599   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
600   if (afill < 1.0) afill = 1.0;
601   c->maxnz                     = ci[am];
602   c->nz                        = ci[am];
603   (*C)->info.mallocs           = ndouble;
604   (*C)->info.fill_ratio_given  = fill;
605   (*C)->info.fill_ratio_needed = afill;
606 
607 #if defined(PETSC_USE_INFO)
608   if (ci[am]) {
609     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
610     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
611   } else {
612     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
613   }
614 #endif
615   PetscFunctionReturn(0);
616 }
617 
618 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_BTHeap(Mat A,Mat B,PetscReal fill,Mat *C)
619 {
620   PetscErrorCode     ierr;
621   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
622   const PetscInt     *ai = a->i,*bi=b->i,*aj=a->j,*bj=b->j;
623   PetscInt           *ci,*cj,*bb;
624   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
625   PetscReal          afill;
626   PetscInt           i,j,col,ndouble = 0;
627   PetscFreeSpaceList free_space=NULL,current_space=NULL;
628   PetscHeap          h;
629   PetscBT            bt;
630 
631   PetscFunctionBegin;
632   /* Get ci and cj - using a heap for the sorted rows, but use BT so that each index is only added once */
633   /*---------------------------------------------------------------------------------------------*/
634   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
635   ierr  = PetscMalloc1(am+2,&ci);CHKERRQ(ierr);
636   ci[0] = 0;
637 
638   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
639   ierr = PetscFreeSpaceGet(PetscRealIntMultTruncate(fill,PetscIntSumTruncate(ai[am],bi[bm])),&free_space);CHKERRQ(ierr);
640 
641   current_space = free_space;
642 
643   ierr = PetscHeapCreate(a->rmax,&h);CHKERRQ(ierr);
644   ierr = PetscMalloc1(a->rmax,&bb);CHKERRQ(ierr);
645   ierr = PetscBTCreate(bn,&bt);CHKERRQ(ierr);
646 
647   /* Determine ci and cj */
648   for (i=0; i<am; i++) {
649     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 */
650     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
651     const PetscInt *fptr = current_space->array; /* Save beginning of the row so we can clear the BT later */
652     ci[i+1] = ci[i];
653     /* Populate the min heap */
654     for (j=0; j<anzi; j++) {
655       PetscInt brow = acol[j];
656       for (bb[j] = bi[brow]; bb[j] < bi[brow+1]; bb[j]++) {
657         PetscInt bcol = bj[bb[j]];
658         if (!PetscBTLookupSet(bt,bcol)) { /* new entry */
659           ierr = PetscHeapAdd(h,j,bcol);CHKERRQ(ierr);
660           bb[j]++;
661           break;
662         }
663       }
664     }
665     /* Pick off the min element, adding it to free space */
666     ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
667     while (j >= 0) {
668       if (current_space->local_remaining < 1) { /* double the size, but don't exceed 16 MiB */
669         fptr = NULL;                      /* need PetscBTMemzero */
670         ierr = PetscFreeSpaceGet(PetscMin(PetscIntMultTruncate(2,current_space->total_array_size),16 << 20),&current_space);CHKERRQ(ierr);
671         ndouble++;
672       }
673       *(current_space->array++) = col;
674       current_space->local_used++;
675       current_space->local_remaining--;
676       ci[i+1]++;
677 
678       /* stash if anything else remains in this row of B */
679       for (; bb[j] < bi[acol[j]+1]; bb[j]++) {
680         PetscInt bcol = bj[bb[j]];
681         if (!PetscBTLookupSet(bt,bcol)) { /* new entry */
682           ierr = PetscHeapAdd(h,j,bcol);CHKERRQ(ierr);
683           bb[j]++;
684           break;
685         }
686       }
687       ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
688     }
689     if (fptr) {                 /* Clear the bits for this row */
690       for (; fptr<current_space->array; fptr++) {ierr = PetscBTClear(bt,*fptr);CHKERRQ(ierr);}
691     } else {                    /* We reallocated so we don't remember (easily) how to clear only the bits we changed */
692       ierr = PetscBTMemzero(bn,bt);CHKERRQ(ierr);
693     }
694   }
695   ierr = PetscFree(bb);CHKERRQ(ierr);
696   ierr = PetscHeapDestroy(&h);CHKERRQ(ierr);
697   ierr = PetscBTDestroy(&bt);CHKERRQ(ierr);
698 
699   /* Column indices are in the list of free space */
700   /* Allocate space for cj, initialize cj, and */
701   /* destroy list of free space and other temporary array(s) */
702   ierr = PetscMalloc1(ci[am],&cj);CHKERRQ(ierr);
703   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
704 
705   /* put together the new symbolic matrix */
706   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
707   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
708   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
709 
710   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
711   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
712   c          = (Mat_SeqAIJ*)((*C)->data);
713   c->free_a  = PETSC_TRUE;
714   c->free_ij = PETSC_TRUE;
715   c->nonew   = 0;
716 
717   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ;
718 
719   /* set MatInfo */
720   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
721   if (afill < 1.0) afill = 1.0;
722   c->maxnz                     = ci[am];
723   c->nz                        = ci[am];
724   (*C)->info.mallocs           = ndouble;
725   (*C)->info.fill_ratio_given  = fill;
726   (*C)->info.fill_ratio_needed = afill;
727 
728 #if defined(PETSC_USE_INFO)
729   if (ci[am]) {
730     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
731     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
732   } else {
733     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
734   }
735 #endif
736   PetscFunctionReturn(0);
737 }
738 
739 /* concatenate unique entries and then sort */
740 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat *C)
741 {
742   PetscErrorCode     ierr;
743   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
744   const PetscInt     *ai = a->i,*bi=b->i,*aj=a->j,*bj=b->j;
745   PetscInt           *ci,*cj;
746   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
747   PetscReal          afill;
748   PetscInt           i,j,ndouble = 0;
749   PetscSegBuffer     seg,segrow;
750   char               *seen;
751 
752   PetscFunctionBegin;
753   ierr  = PetscMalloc1(am+1,&ci);CHKERRQ(ierr);
754   ci[0] = 0;
755 
756   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
757   ierr = PetscSegBufferCreate(sizeof(PetscInt),(PetscInt)(fill*(ai[am]+bi[bm])),&seg);CHKERRQ(ierr);
758   ierr = PetscSegBufferCreate(sizeof(PetscInt),100,&segrow);CHKERRQ(ierr);
759   ierr = PetscMalloc1(bn,&seen);CHKERRQ(ierr);
760   ierr = PetscMemzero(seen,bn*sizeof(char));CHKERRQ(ierr);
761 
762   /* Determine ci and cj */
763   for (i=0; i<am; i++) {
764     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 */
765     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
766     PetscInt packlen = 0,*PETSC_RESTRICT crow;
767     /* Pack segrow */
768     for (j=0; j<anzi; j++) {
769       PetscInt brow = acol[j],bjstart = bi[brow],bjend = bi[brow+1],k;
770       for (k=bjstart; k<bjend; k++) {
771         PetscInt bcol = bj[k];
772         if (!seen[bcol]) { /* new entry */
773           PetscInt *PETSC_RESTRICT slot;
774           ierr = PetscSegBufferGetInts(segrow,1,&slot);CHKERRQ(ierr);
775           *slot = bcol;
776           seen[bcol] = 1;
777           packlen++;
778         }
779       }
780     }
781     ierr = PetscSegBufferGetInts(seg,packlen,&crow);CHKERRQ(ierr);
782     ierr = PetscSegBufferExtractTo(segrow,crow);CHKERRQ(ierr);
783     ierr = PetscSortInt(packlen,crow);CHKERRQ(ierr);
784     ci[i+1] = ci[i] + packlen;
785     for (j=0; j<packlen; j++) seen[crow[j]] = 0;
786   }
787   ierr = PetscSegBufferDestroy(&segrow);CHKERRQ(ierr);
788   ierr = PetscFree(seen);CHKERRQ(ierr);
789 
790   /* Column indices are in the segmented buffer */
791   ierr = PetscSegBufferExtractAlloc(seg,&cj);CHKERRQ(ierr);
792   ierr = PetscSegBufferDestroy(&seg);CHKERRQ(ierr);
793 
794   /* put together the new symbolic matrix */
795   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
796   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
797   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
798 
799   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
800   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
801   c          = (Mat_SeqAIJ*)((*C)->data);
802   c->free_a  = PETSC_TRUE;
803   c->free_ij = PETSC_TRUE;
804   c->nonew   = 0;
805 
806   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ;
807 
808   /* set MatInfo */
809   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
810   if (afill < 1.0) afill = 1.0;
811   c->maxnz                     = ci[am];
812   c->nz                        = ci[am];
813   (*C)->info.mallocs           = ndouble;
814   (*C)->info.fill_ratio_given  = fill;
815   (*C)->info.fill_ratio_needed = afill;
816 
817 #if defined(PETSC_USE_INFO)
818   if (ci[am]) {
819     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %g needed %g.\n",ndouble,(double)fill,(double)afill);CHKERRQ(ierr);
820     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%g,&C) for best performance.;\n",(double)afill);CHKERRQ(ierr);
821   } else {
822     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
823   }
824 #endif
825   PetscFunctionReturn(0);
826 }
827 
828 /* This routine is not used. Should be removed! */
829 PetscErrorCode MatMatTransposeMult_SeqAIJ_SeqAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
830 {
831   PetscErrorCode ierr;
832 
833   PetscFunctionBegin;
834   if (scall == MAT_INITIAL_MATRIX) {
835     ierr = PetscLogEventBegin(MAT_MatTransposeMultSymbolic,A,B,0,0);CHKERRQ(ierr);
836     ierr = MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ(A,B,fill,C);CHKERRQ(ierr);
837     ierr = PetscLogEventEnd(MAT_MatTransposeMultSymbolic,A,B,0,0);CHKERRQ(ierr);
838   }
839   ierr = PetscLogEventBegin(MAT_MatTransposeMultNumeric,A,B,0,0);CHKERRQ(ierr);
840   ierr = MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ(A,B,*C);CHKERRQ(ierr);
841   ierr = PetscLogEventEnd(MAT_MatTransposeMultNumeric,A,B,0,0);CHKERRQ(ierr);
842   PetscFunctionReturn(0);
843 }
844 
845 PetscErrorCode MatDestroy_SeqAIJ_MatMatMultTrans(Mat A)
846 {
847   PetscErrorCode      ierr;
848   Mat_SeqAIJ          *a=(Mat_SeqAIJ*)A->data;
849   Mat_MatMatTransMult *abt=a->abt;
850 
851   PetscFunctionBegin;
852   ierr = (abt->destroy)(A);CHKERRQ(ierr);
853   ierr = MatTransposeColoringDestroy(&abt->matcoloring);CHKERRQ(ierr);
854   ierr = MatDestroy(&abt->Bt_den);CHKERRQ(ierr);
855   ierr = MatDestroy(&abt->ABt_den);CHKERRQ(ierr);
856   ierr = PetscFree(abt);CHKERRQ(ierr);
857   PetscFunctionReturn(0);
858 }
859 
860 PetscErrorCode MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat *C)
861 {
862   PetscErrorCode      ierr;
863   Mat                 Bt;
864   PetscInt            *bti,*btj;
865   Mat_MatMatTransMult *abt;
866   Mat_SeqAIJ          *c;
867 
868   PetscFunctionBegin;
869   /* create symbolic Bt */
870   ierr = MatGetSymbolicTranspose_SeqAIJ(B,&bti,&btj);CHKERRQ(ierr);
871   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,B->cmap->n,B->rmap->n,bti,btj,NULL,&Bt);CHKERRQ(ierr);
872   ierr = MatSetBlockSizes(Bt,PetscAbs(A->cmap->bs),PetscAbs(B->cmap->bs));CHKERRQ(ierr);
873   ierr = MatSetType(Bt,((PetscObject)A)->type_name);CHKERRQ(ierr);
874 
875   /* get symbolic C=A*Bt */
876   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(A,Bt,fill,C);CHKERRQ(ierr);
877 
878   /* create a supporting struct for reuse intermidiate dense matrices with matcoloring */
879   ierr   = PetscNew(&abt);CHKERRQ(ierr);
880   c      = (Mat_SeqAIJ*)(*C)->data;
881   c->abt = abt;
882 
883   abt->usecoloring = PETSC_FALSE;
884   abt->destroy     = (*C)->ops->destroy;
885   (*C)->ops->destroy     = MatDestroy_SeqAIJ_MatMatMultTrans;
886 
887   ierr = PetscOptionsGetBool(((PetscObject)A)->options,NULL,"-matmattransmult_color",&abt->usecoloring,NULL);CHKERRQ(ierr);
888   if (abt->usecoloring) {
889     /* Create MatTransposeColoring from symbolic C=A*B^T */
890     MatTransposeColoring matcoloring;
891     MatColoring          coloring;
892     ISColoring           iscoloring;
893     Mat                  Bt_dense,C_dense;
894     Mat_SeqAIJ           *c=(Mat_SeqAIJ*)(*C)->data;
895     /* inode causes memory problem, don't know why */
896     if (c->inode.use) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MAT_USE_INODES is not supported. Use '-mat_no_inode'");
897 
898     ierr = MatColoringCreate(*C,&coloring);CHKERRQ(ierr);
899     ierr = MatColoringSetDistance(coloring,2);CHKERRQ(ierr);
900     ierr = MatColoringSetType(coloring,MATCOLORINGSL);CHKERRQ(ierr);
901     ierr = MatColoringSetFromOptions(coloring);CHKERRQ(ierr);
902     ierr = MatColoringApply(coloring,&iscoloring);CHKERRQ(ierr);
903     ierr = MatColoringDestroy(&coloring);CHKERRQ(ierr);
904     ierr = MatTransposeColoringCreate(*C,iscoloring,&matcoloring);CHKERRQ(ierr);
905 
906     abt->matcoloring = matcoloring;
907 
908     ierr = ISColoringDestroy(&iscoloring);CHKERRQ(ierr);
909 
910     /* Create Bt_dense and C_dense = A*Bt_dense */
911     ierr = MatCreate(PETSC_COMM_SELF,&Bt_dense);CHKERRQ(ierr);
912     ierr = MatSetSizes(Bt_dense,A->cmap->n,matcoloring->ncolors,A->cmap->n,matcoloring->ncolors);CHKERRQ(ierr);
913     ierr = MatSetType(Bt_dense,MATSEQDENSE);CHKERRQ(ierr);
914     ierr = MatSeqDenseSetPreallocation(Bt_dense,NULL);CHKERRQ(ierr);
915 
916     Bt_dense->assembled = PETSC_TRUE;
917     abt->Bt_den   = Bt_dense;
918 
919     ierr = MatCreate(PETSC_COMM_SELF,&C_dense);CHKERRQ(ierr);
920     ierr = MatSetSizes(C_dense,A->rmap->n,matcoloring->ncolors,A->rmap->n,matcoloring->ncolors);CHKERRQ(ierr);
921     ierr = MatSetType(C_dense,MATSEQDENSE);CHKERRQ(ierr);
922     ierr = MatSeqDenseSetPreallocation(C_dense,NULL);CHKERRQ(ierr);
923 
924     Bt_dense->assembled = PETSC_TRUE;
925     abt->ABt_den  = C_dense;
926 
927 #if defined(PETSC_USE_INFO)
928     {
929       Mat_SeqAIJ *c = (Mat_SeqAIJ*)(*C)->data;
930       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);
931     }
932 #endif
933   }
934   /* clean up */
935   ierr = MatDestroy(&Bt);CHKERRQ(ierr);
936   ierr = MatRestoreSymbolicTranspose_SeqAIJ(B,&bti,&btj);CHKERRQ(ierr);
937   PetscFunctionReturn(0);
938 }
939 
940 PetscErrorCode MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
941 {
942   PetscErrorCode      ierr;
943   Mat_SeqAIJ          *a   =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c=(Mat_SeqAIJ*)C->data;
944   PetscInt            *ai  =a->i,*aj=a->j,*bi=b->i,*bj=b->j,anzi,bnzj,nexta,nextb,*acol,*bcol,brow;
945   PetscInt            cm   =C->rmap->n,*ci=c->i,*cj=c->j,i,j,cnzi,*ccol;
946   PetscLogDouble      flops=0.0;
947   MatScalar           *aa  =a->a,*aval,*ba=b->a,*bval,*ca,*cval;
948   Mat_MatMatTransMult *abt = c->abt;
949 
950   PetscFunctionBegin;
951   /* clear old values in C */
952   if (!c->a) {
953     ierr      = PetscMalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
954     c->a      = ca;
955     c->free_a = PETSC_TRUE;
956   } else {
957     ca =  c->a;
958   }
959   ierr = PetscMemzero(ca,ci[cm]*sizeof(MatScalar));CHKERRQ(ierr);
960 
961   if (abt->usecoloring) {
962     MatTransposeColoring matcoloring = abt->matcoloring;
963     Mat                  Bt_dense,C_dense = abt->ABt_den;
964 
965     /* Get Bt_dense by Apply MatTransposeColoring to B */
966     Bt_dense = abt->Bt_den;
967     ierr = MatTransColoringApplySpToDen(matcoloring,B,Bt_dense);CHKERRQ(ierr);
968 
969     /* C_dense = A*Bt_dense */
970     ierr = MatMatMultNumeric_SeqAIJ_SeqDense(A,Bt_dense,C_dense);CHKERRQ(ierr);
971 
972     /* Recover C from C_dense */
973     ierr = MatTransColoringApplyDenToSp(matcoloring,C_dense,C);CHKERRQ(ierr);
974     PetscFunctionReturn(0);
975   }
976 
977   for (i=0; i<cm; i++) {
978     anzi = ai[i+1] - ai[i];
979     acol = aj + ai[i];
980     aval = aa + ai[i];
981     cnzi = ci[i+1] - ci[i];
982     ccol = cj + ci[i];
983     cval = ca + ci[i];
984     for (j=0; j<cnzi; j++) {
985       brow = ccol[j];
986       bnzj = bi[brow+1] - bi[brow];
987       bcol = bj + bi[brow];
988       bval = ba + bi[brow];
989 
990       /* perform sparse inner-product c(i,j)=A[i,:]*B[j,:]^T */
991       nexta = 0; nextb = 0;
992       while (nexta<anzi && nextb<bnzj) {
993         while (nexta < anzi && acol[nexta] < bcol[nextb]) nexta++;
994         if (nexta == anzi) break;
995         while (nextb < bnzj && acol[nexta] > bcol[nextb]) nextb++;
996         if (nextb == bnzj) break;
997         if (acol[nexta] == bcol[nextb]) {
998           cval[j] += aval[nexta]*bval[nextb];
999           nexta++; nextb++;
1000           flops += 2;
1001         }
1002       }
1003     }
1004   }
1005   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1006   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1007   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
1008   PetscFunctionReturn(0);
1009 }
1010 
1011 PetscErrorCode MatDestroy_SeqAIJ_MatTransMatMult(Mat A)
1012 {
1013   PetscErrorCode      ierr;
1014   Mat_SeqAIJ          *a = (Mat_SeqAIJ*)A->data;
1015   Mat_MatTransMatMult *atb = a->atb;
1016 
1017   PetscFunctionBegin;
1018   ierr = MatDestroy(&atb->At);CHKERRQ(ierr);
1019   ierr = (atb->destroy)(A);CHKERRQ(ierr);
1020   ierr = PetscFree(atb);CHKERRQ(ierr);
1021   PetscFunctionReturn(0);
1022 }
1023 
1024 PetscErrorCode MatTransposeMatMult_SeqAIJ_SeqAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
1025 {
1026   PetscErrorCode      ierr;
1027   const char          *algTypes[2] = {"matmatmult","outerproduct"};
1028   PetscInt            alg=0; /* set default algorithm */
1029   Mat                 At;
1030   Mat_MatTransMatMult *atb;
1031   Mat_SeqAIJ          *c;
1032 
1033   PetscFunctionBegin;
1034   if (scall == MAT_INITIAL_MATRIX) {
1035     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
1036     PetscOptionsObject->alreadyprinted = PETSC_FALSE; /* a hack to ensure the option shows in '-help' */
1037     ierr = PetscOptionsEList("-mattransposematmult_via","Algorithmic approach","MatTransposeMatMult",algTypes,2,algTypes[0],&alg,NULL);CHKERRQ(ierr);
1038     ierr = PetscOptionsEnd();CHKERRQ(ierr);
1039 
1040     switch (alg) {
1041     case 1:
1042       ierr = MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ(A,B,fill,C);CHKERRQ(ierr);
1043       break;
1044     default:
1045       ierr = PetscNew(&atb);CHKERRQ(ierr);
1046       ierr = MatTranspose_SeqAIJ(A,MAT_INITIAL_MATRIX,&At);CHKERRQ(ierr);
1047       ierr = MatMatMult_SeqAIJ_SeqAIJ(At,B,MAT_INITIAL_MATRIX,fill,C);CHKERRQ(ierr);
1048 
1049       c                  = (Mat_SeqAIJ*)(*C)->data;
1050       c->atb             = atb;
1051       atb->At            = At;
1052       atb->destroy       = (*C)->ops->destroy;
1053       (*C)->ops->destroy = MatDestroy_SeqAIJ_MatTransMatMult;
1054 
1055       break;
1056     }
1057   }
1058   if (alg) {
1059     ierr = (*(*C)->ops->mattransposemultnumeric)(A,B,*C);CHKERRQ(ierr);
1060   } else if (!alg && scall == MAT_REUSE_MATRIX) {
1061     c   = (Mat_SeqAIJ*)(*C)->data;
1062     atb = c->atb;
1063     At  = atb->At;
1064     ierr = MatTranspose_SeqAIJ(A,MAT_REUSE_MATRIX,&At);CHKERRQ(ierr);
1065     ierr = MatMatMult_SeqAIJ_SeqAIJ(At,B,MAT_REUSE_MATRIX,fill,C);CHKERRQ(ierr);
1066   }
1067   PetscFunctionReturn(0);
1068 }
1069 
1070 PetscErrorCode MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat *C)
1071 {
1072   PetscErrorCode ierr;
1073   Mat            At;
1074   PetscInt       *ati,*atj;
1075 
1076   PetscFunctionBegin;
1077   /* create symbolic At */
1078   ierr = MatGetSymbolicTranspose_SeqAIJ(A,&ati,&atj);CHKERRQ(ierr);
1079   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,A->cmap->n,A->rmap->n,ati,atj,NULL,&At);CHKERRQ(ierr);
1080   ierr = MatSetBlockSizes(At,PetscAbs(A->cmap->bs),PetscAbs(B->cmap->bs));CHKERRQ(ierr);
1081   ierr = MatSetType(At,((PetscObject)A)->type_name);CHKERRQ(ierr);
1082 
1083   /* get symbolic C=At*B */
1084   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(At,B,fill,C);CHKERRQ(ierr);
1085 
1086   /* clean up */
1087   ierr = MatDestroy(&At);CHKERRQ(ierr);
1088   ierr = MatRestoreSymbolicTranspose_SeqAIJ(A,&ati,&atj);CHKERRQ(ierr);
1089 
1090   (*C)->ops->mattransposemultnumeric = MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ;
1091   PetscFunctionReturn(0);
1092 }
1093 
1094 PetscErrorCode MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
1095 {
1096   PetscErrorCode ierr;
1097   Mat_SeqAIJ     *a   =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c=(Mat_SeqAIJ*)C->data;
1098   PetscInt       am   =A->rmap->n,anzi,*ai=a->i,*aj=a->j,*bi=b->i,*bj,bnzi,nextb;
1099   PetscInt       cm   =C->rmap->n,*ci=c->i,*cj=c->j,crow,*cjj,i,j,k;
1100   PetscLogDouble flops=0.0;
1101   MatScalar      *aa  =a->a,*ba,*ca,*caj;
1102 
1103   PetscFunctionBegin;
1104   if (!c->a) {
1105     ierr = PetscMalloc1(ci[cm]+1,&ca);CHKERRQ(ierr);
1106 
1107     c->a      = ca;
1108     c->free_a = PETSC_TRUE;
1109   } else {
1110     ca = c->a;
1111   }
1112   /* clear old values in C */
1113   ierr = PetscMemzero(ca,ci[cm]*sizeof(MatScalar));CHKERRQ(ierr);
1114 
1115   /* compute A^T*B using outer product (A^T)[:,i]*B[i,:] */
1116   for (i=0; i<am; i++) {
1117     bj   = b->j + bi[i];
1118     ba   = b->a + bi[i];
1119     bnzi = bi[i+1] - bi[i];
1120     anzi = ai[i+1] - ai[i];
1121     for (j=0; j<anzi; j++) {
1122       nextb = 0;
1123       crow  = *aj++;
1124       cjj   = cj + ci[crow];
1125       caj   = ca + ci[crow];
1126       /* perform sparse axpy operation.  Note cjj includes bj. */
1127       for (k=0; nextb<bnzi; k++) {
1128         if (cjj[k] == *(bj+nextb)) { /* ccol == bcol */
1129           caj[k] += (*aa)*(*(ba+nextb));
1130           nextb++;
1131         }
1132       }
1133       flops += 2*bnzi;
1134       aa++;
1135     }
1136   }
1137 
1138   /* Assemble the final matrix and clean up */
1139   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1140   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1141   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
1142   PetscFunctionReturn(0);
1143 }
1144 
1145 PETSC_INTERN PetscErrorCode MatMatMult_SeqAIJ_SeqDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
1146 {
1147   PetscErrorCode ierr;
1148 
1149   PetscFunctionBegin;
1150   if (scall == MAT_INITIAL_MATRIX) {
1151     ierr = PetscLogEventBegin(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
1152     ierr = MatMatMultSymbolic_SeqAIJ_SeqDense(A,B,fill,C);CHKERRQ(ierr);
1153     ierr = PetscLogEventEnd(MAT_MatMultSymbolic,A,B,0,0);CHKERRQ(ierr);
1154   }
1155   ierr = PetscLogEventBegin(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
1156   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(A,B,*C);CHKERRQ(ierr);
1157   ierr = PetscLogEventEnd(MAT_MatMultNumeric,A,B,0,0);CHKERRQ(ierr);
1158   PetscFunctionReturn(0);
1159 }
1160 
1161 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqDense(Mat A,Mat B,PetscReal fill,Mat *C)
1162 {
1163   PetscErrorCode ierr;
1164 
1165   PetscFunctionBegin;
1166   ierr = MatMatMultSymbolic_SeqDense_SeqDense(A,B,0.0,C);CHKERRQ(ierr);
1167 
1168   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqDense;
1169   PetscFunctionReturn(0);
1170 }
1171 
1172 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqDense(Mat A,Mat B,Mat C)
1173 {
1174   Mat_SeqAIJ        *a=(Mat_SeqAIJ*)A->data;
1175   Mat_SeqDense      *bd = (Mat_SeqDense*)B->data;
1176   PetscErrorCode    ierr;
1177   PetscScalar       *c,*b,r1,r2,r3,r4,*c1,*c2,*c3,*c4,aatmp;
1178   const PetscScalar *aa,*b1,*b2,*b3,*b4;
1179   const PetscInt    *aj;
1180   PetscInt          cm=C->rmap->n,cn=B->cmap->n,bm=bd->lda,am=A->rmap->n;
1181   PetscInt          am4=4*am,bm4=4*bm,col,i,j,n,ajtmp;
1182 
1183   PetscFunctionBegin;
1184   if (!cm || !cn) PetscFunctionReturn(0);
1185   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);
1186   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);
1187   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);
1188   b = bd->v;
1189   ierr = MatDenseGetArray(C,&c);CHKERRQ(ierr);
1190   b1 = b; b2 = b1 + bm; b3 = b2 + bm; b4 = b3 + bm;
1191   c1 = c; c2 = c1 + am; c3 = c2 + am; c4 = c3 + am;
1192   for (col=0; col<cn-4; col += 4) {  /* over columns of C */
1193     for (i=0; i<am; i++) {        /* over rows of C in those columns */
1194       r1 = r2 = r3 = r4 = 0.0;
1195       n  = a->i[i+1] - a->i[i];
1196       aj = a->j + a->i[i];
1197       aa = a->a + a->i[i];
1198       for (j=0; j<n; j++) {
1199         aatmp = aa[j]; ajtmp = aj[j];
1200         r1 += aatmp*b1[ajtmp];
1201         r2 += aatmp*b2[ajtmp];
1202         r3 += aatmp*b3[ajtmp];
1203         r4 += aatmp*b4[ajtmp];
1204       }
1205       c1[i] = r1;
1206       c2[i] = r2;
1207       c3[i] = r3;
1208       c4[i] = r4;
1209     }
1210     b1 += bm4; b2 += bm4; b3 += bm4; b4 += bm4;
1211     c1 += am4; c2 += am4; c3 += am4; c4 += am4;
1212   }
1213   for (; col<cn; col++) {   /* over extra columns of C */
1214     for (i=0; i<am; i++) {  /* over rows of C in those columns */
1215       r1 = 0.0;
1216       n  = a->i[i+1] - a->i[i];
1217       aj = a->j + a->i[i];
1218       aa = a->a + a->i[i];
1219       for (j=0; j<n; j++) {
1220         r1 += aa[j]*b1[aj[j]];
1221       }
1222       c1[i] = r1;
1223     }
1224     b1 += bm;
1225     c1 += am;
1226   }
1227   ierr = PetscLogFlops(cn*(2.0*a->nz));CHKERRQ(ierr);
1228   ierr = MatDenseRestoreArray(C,&c);CHKERRQ(ierr);
1229   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1230   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1231   PetscFunctionReturn(0);
1232 }
1233 
1234 /*
1235    Note very similar to MatMult_SeqAIJ(), should generate both codes from same base
1236 */
1237 PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat A,Mat B,Mat C)
1238 {
1239   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
1240   Mat_SeqDense   *bd = (Mat_SeqDense*)B->data;
1241   PetscErrorCode ierr;
1242   PetscScalar    *b,*c,r1,r2,r3,r4,*b1,*b2,*b3,*b4;
1243   MatScalar      *aa;
1244   PetscInt       cm  = C->rmap->n, cn=B->cmap->n, bm=bd->lda, col, i,j,n,*aj, am = A->rmap->n,*ii,arm;
1245   PetscInt       am2 = 2*am, am3 = 3*am,  bm4 = 4*bm,colam,*ridx;
1246 
1247   PetscFunctionBegin;
1248   if (!cm || !cn) PetscFunctionReturn(0);
1249   b = bd->v;
1250   ierr = MatDenseGetArray(C,&c);CHKERRQ(ierr);
1251   b1   = b; b2 = b1 + bm; b3 = b2 + bm; b4 = b3 + bm;
1252 
1253   if (a->compressedrow.use) { /* use compressed row format */
1254     for (col=0; col<cn-4; col += 4) {  /* over columns of C */
1255       colam = col*am;
1256       arm   = a->compressedrow.nrows;
1257       ii    = a->compressedrow.i;
1258       ridx  = a->compressedrow.rindex;
1259       for (i=0; i<arm; i++) {        /* over rows of C in those columns */
1260         r1 = r2 = r3 = r4 = 0.0;
1261         n  = ii[i+1] - ii[i];
1262         aj = a->j + ii[i];
1263         aa = a->a + ii[i];
1264         for (j=0; j<n; j++) {
1265           r1 += (*aa)*b1[*aj];
1266           r2 += (*aa)*b2[*aj];
1267           r3 += (*aa)*b3[*aj];
1268           r4 += (*aa++)*b4[*aj++];
1269         }
1270         c[colam       + ridx[i]] += r1;
1271         c[colam + am  + ridx[i]] += r2;
1272         c[colam + am2 + ridx[i]] += r3;
1273         c[colam + am3 + ridx[i]] += r4;
1274       }
1275       b1 += bm4;
1276       b2 += bm4;
1277       b3 += bm4;
1278       b4 += bm4;
1279     }
1280     for (; col<cn; col++) {     /* over extra columns of C */
1281       colam = col*am;
1282       arm   = a->compressedrow.nrows;
1283       ii    = a->compressedrow.i;
1284       ridx  = a->compressedrow.rindex;
1285       for (i=0; i<arm; i++) {  /* over rows of C in those columns */
1286         r1 = 0.0;
1287         n  = ii[i+1] - ii[i];
1288         aj = a->j + ii[i];
1289         aa = a->a + ii[i];
1290 
1291         for (j=0; j<n; j++) {
1292           r1 += (*aa++)*b1[*aj++];
1293         }
1294         c[colam + ridx[i]] += r1;
1295       }
1296       b1 += bm;
1297     }
1298   } else {
1299     for (col=0; col<cn-4; col += 4) {  /* over columns of C */
1300       colam = col*am;
1301       for (i=0; i<am; i++) {        /* over rows of C in those columns */
1302         r1 = r2 = r3 = r4 = 0.0;
1303         n  = a->i[i+1] - a->i[i];
1304         aj = a->j + a->i[i];
1305         aa = a->a + a->i[i];
1306         for (j=0; j<n; j++) {
1307           r1 += (*aa)*b1[*aj];
1308           r2 += (*aa)*b2[*aj];
1309           r3 += (*aa)*b3[*aj];
1310           r4 += (*aa++)*b4[*aj++];
1311         }
1312         c[colam + i]       += r1;
1313         c[colam + am + i]  += r2;
1314         c[colam + am2 + i] += r3;
1315         c[colam + am3 + i] += r4;
1316       }
1317       b1 += bm4;
1318       b2 += bm4;
1319       b3 += bm4;
1320       b4 += bm4;
1321     }
1322     for (; col<cn; col++) {     /* over extra columns of C */
1323       colam = col*am;
1324       for (i=0; i<am; i++) {  /* over rows of C in those columns */
1325         r1 = 0.0;
1326         n  = a->i[i+1] - a->i[i];
1327         aj = a->j + a->i[i];
1328         aa = a->a + a->i[i];
1329 
1330         for (j=0; j<n; j++) {
1331           r1 += (*aa++)*b1[*aj++];
1332         }
1333         c[colam + i] += r1;
1334       }
1335       b1 += bm;
1336     }
1337   }
1338   ierr = PetscLogFlops(cn*2.0*a->nz);CHKERRQ(ierr);
1339   ierr = MatDenseRestoreArray(C,&c);CHKERRQ(ierr);
1340   PetscFunctionReturn(0);
1341 }
1342 
1343 PetscErrorCode  MatTransColoringApplySpToDen_SeqAIJ(MatTransposeColoring coloring,Mat B,Mat Btdense)
1344 {
1345   PetscErrorCode ierr;
1346   Mat_SeqAIJ     *b       = (Mat_SeqAIJ*)B->data;
1347   Mat_SeqDense   *btdense = (Mat_SeqDense*)Btdense->data;
1348   PetscInt       *bi      = b->i,*bj=b->j;
1349   PetscInt       m        = Btdense->rmap->n,n=Btdense->cmap->n,j,k,l,col,anz,*btcol,brow,ncolumns;
1350   MatScalar      *btval,*btval_den,*ba=b->a;
1351   PetscInt       *columns=coloring->columns,*colorforcol=coloring->colorforcol,ncolors=coloring->ncolors;
1352 
1353   PetscFunctionBegin;
1354   btval_den=btdense->v;
1355   ierr     = PetscMemzero(btval_den,(m*n)*sizeof(MatScalar));CHKERRQ(ierr);
1356   for (k=0; k<ncolors; k++) {
1357     ncolumns = coloring->ncolumns[k];
1358     for (l=0; l<ncolumns; l++) { /* insert a row of B to a column of Btdense */
1359       col   = *(columns + colorforcol[k] + l);
1360       btcol = bj + bi[col];
1361       btval = ba + bi[col];
1362       anz   = bi[col+1] - bi[col];
1363       for (j=0; j<anz; j++) {
1364         brow            = btcol[j];
1365         btval_den[brow] = btval[j];
1366       }
1367     }
1368     btval_den += m;
1369   }
1370   PetscFunctionReturn(0);
1371 }
1372 
1373 PetscErrorCode MatTransColoringApplyDenToSp_SeqAIJ(MatTransposeColoring matcoloring,Mat Cden,Mat Csp)
1374 {
1375   PetscErrorCode ierr;
1376   Mat_SeqAIJ     *csp = (Mat_SeqAIJ*)Csp->data;
1377   PetscScalar    *ca_den,*ca_den_ptr,*ca=csp->a;
1378   PetscInt       k,l,m=Cden->rmap->n,ncolors=matcoloring->ncolors;
1379   PetscInt       brows=matcoloring->brows,*den2sp=matcoloring->den2sp;
1380   PetscInt       nrows,*row,*idx;
1381   PetscInt       *rows=matcoloring->rows,*colorforrow=matcoloring->colorforrow;
1382 
1383   PetscFunctionBegin;
1384   ierr   = MatDenseGetArray(Cden,&ca_den);CHKERRQ(ierr);
1385 
1386   if (brows > 0) {
1387     PetscInt *lstart,row_end,row_start;
1388     lstart = matcoloring->lstart;
1389     ierr = PetscMemzero(lstart,ncolors*sizeof(PetscInt));CHKERRQ(ierr);
1390 
1391     row_end = brows;
1392     if (row_end > m) row_end = m;
1393     for (row_start=0; row_start<m; row_start+=brows) { /* loop over row blocks of Csp */
1394       ca_den_ptr = ca_den;
1395       for (k=0; k<ncolors; k++) { /* loop over colors (columns of Cden) */
1396         nrows = matcoloring->nrows[k];
1397         row   = rows  + colorforrow[k];
1398         idx   = den2sp + colorforrow[k];
1399         for (l=lstart[k]; l<nrows; l++) {
1400           if (row[l] >= row_end) {
1401             lstart[k] = l;
1402             break;
1403           } else {
1404             ca[idx[l]] = ca_den_ptr[row[l]];
1405           }
1406         }
1407         ca_den_ptr += m;
1408       }
1409       row_end += brows;
1410       if (row_end > m) row_end = m;
1411     }
1412   } else { /* non-blocked impl: loop over columns of Csp - slow if Csp is large */
1413     ca_den_ptr = ca_den;
1414     for (k=0; k<ncolors; k++) {
1415       nrows = matcoloring->nrows[k];
1416       row   = rows  + colorforrow[k];
1417       idx   = den2sp + colorforrow[k];
1418       for (l=0; l<nrows; l++) {
1419         ca[idx[l]] = ca_den_ptr[row[l]];
1420       }
1421       ca_den_ptr += m;
1422     }
1423   }
1424 
1425   ierr = MatDenseRestoreArray(Cden,&ca_den);CHKERRQ(ierr);
1426 #if defined(PETSC_USE_INFO)
1427   if (matcoloring->brows > 0) {
1428     ierr = PetscInfo1(Csp,"Loop over %D row blocks for den2sp\n",brows);CHKERRQ(ierr);
1429   } else {
1430     ierr = PetscInfo(Csp,"Loop over colors/columns of Cden, inefficient for large sparse matrix product \n");CHKERRQ(ierr);
1431   }
1432 #endif
1433   PetscFunctionReturn(0);
1434 }
1435 
1436 PetscErrorCode MatTransposeColoringCreate_SeqAIJ(Mat mat,ISColoring iscoloring,MatTransposeColoring c)
1437 {
1438   PetscErrorCode ierr;
1439   PetscInt       i,n,nrows,Nbs,j,k,m,ncols,col,cm;
1440   const PetscInt *is,*ci,*cj,*row_idx;
1441   PetscInt       nis = iscoloring->n,*rowhit,bs = 1;
1442   IS             *isa;
1443   Mat_SeqAIJ     *csp = (Mat_SeqAIJ*)mat->data;
1444   PetscInt       *colorforrow,*rows,*rows_i,*idxhit,*spidx,*den2sp,*den2sp_i;
1445   PetscInt       *colorforcol,*columns,*columns_i,brows;
1446   PetscBool      flg;
1447 
1448   PetscFunctionBegin;
1449   ierr = ISColoringGetIS(iscoloring,PETSC_IGNORE,&isa);CHKERRQ(ierr);
1450 
1451   /* bs >1 is not being tested yet! */
1452   Nbs       = mat->cmap->N/bs;
1453   c->M      = mat->rmap->N/bs;  /* set total rows, columns and local rows */
1454   c->N      = Nbs;
1455   c->m      = c->M;
1456   c->rstart = 0;
1457   c->brows  = 100;
1458 
1459   c->ncolors = nis;
1460   ierr = PetscMalloc3(nis,&c->ncolumns,nis,&c->nrows,nis+1,&colorforrow);CHKERRQ(ierr);
1461   ierr = PetscMalloc1(csp->nz+1,&rows);CHKERRQ(ierr);
1462   ierr = PetscMalloc1(csp->nz+1,&den2sp);CHKERRQ(ierr);
1463 
1464   brows = c->brows;
1465   ierr = PetscOptionsGetInt(NULL,NULL,"-matden2sp_brows",&brows,&flg);CHKERRQ(ierr);
1466   if (flg) c->brows = brows;
1467   if (brows > 0) {
1468     ierr = PetscMalloc1(nis+1,&c->lstart);CHKERRQ(ierr);
1469   }
1470 
1471   colorforrow[0] = 0;
1472   rows_i         = rows;
1473   den2sp_i       = den2sp;
1474 
1475   ierr = PetscMalloc1(nis+1,&colorforcol);CHKERRQ(ierr);
1476   ierr = PetscMalloc1(Nbs+1,&columns);CHKERRQ(ierr);
1477 
1478   colorforcol[0] = 0;
1479   columns_i      = columns;
1480 
1481   /* get column-wise storage of mat */
1482   ierr = MatGetColumnIJ_SeqAIJ_Color(mat,0,PETSC_FALSE,PETSC_FALSE,&ncols,&ci,&cj,&spidx,NULL);CHKERRQ(ierr);
1483 
1484   cm   = c->m;
1485   ierr = PetscMalloc1(cm+1,&rowhit);CHKERRQ(ierr);
1486   ierr = PetscMalloc1(cm+1,&idxhit);CHKERRQ(ierr);
1487   for (i=0; i<nis; i++) { /* loop over color */
1488     ierr = ISGetLocalSize(isa[i],&n);CHKERRQ(ierr);
1489     ierr = ISGetIndices(isa[i],&is);CHKERRQ(ierr);
1490 
1491     c->ncolumns[i] = n;
1492     if (n) {
1493       ierr = PetscMemcpy(columns_i,is,n*sizeof(PetscInt));CHKERRQ(ierr);
1494     }
1495     colorforcol[i+1] = colorforcol[i] + n;
1496     columns_i       += n;
1497 
1498     /* fast, crude version requires O(N*N) work */
1499     ierr = PetscMemzero(rowhit,cm*sizeof(PetscInt));CHKERRQ(ierr);
1500 
1501     for (j=0; j<n; j++) { /* loop over columns*/
1502       col     = is[j];
1503       row_idx = cj + ci[col];
1504       m       = ci[col+1] - ci[col];
1505       for (k=0; k<m; k++) { /* loop over columns marking them in rowhit */
1506         idxhit[*row_idx]   = spidx[ci[col] + k];
1507         rowhit[*row_idx++] = col + 1;
1508       }
1509     }
1510     /* count the number of hits */
1511     nrows = 0;
1512     for (j=0; j<cm; j++) {
1513       if (rowhit[j]) nrows++;
1514     }
1515     c->nrows[i]      = nrows;
1516     colorforrow[i+1] = colorforrow[i] + nrows;
1517 
1518     nrows = 0;
1519     for (j=0; j<cm; j++) { /* loop over rows */
1520       if (rowhit[j]) {
1521         rows_i[nrows]   = j;
1522         den2sp_i[nrows] = idxhit[j];
1523         nrows++;
1524       }
1525     }
1526     den2sp_i += nrows;
1527 
1528     ierr    = ISRestoreIndices(isa[i],&is);CHKERRQ(ierr);
1529     rows_i += nrows;
1530   }
1531   ierr = MatRestoreColumnIJ_SeqAIJ_Color(mat,0,PETSC_FALSE,PETSC_FALSE,&ncols,&ci,&cj,&spidx,NULL);CHKERRQ(ierr);
1532   ierr = PetscFree(rowhit);CHKERRQ(ierr);
1533   ierr = ISColoringRestoreIS(iscoloring,&isa);CHKERRQ(ierr);
1534   if (csp->nz != colorforrow[nis]) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_PLIB,"csp->nz %d != colorforrow[nis] %d",csp->nz,colorforrow[nis]);
1535 
1536   c->colorforrow = colorforrow;
1537   c->rows        = rows;
1538   c->den2sp      = den2sp;
1539   c->colorforcol = colorforcol;
1540   c->columns     = columns;
1541 
1542   ierr = PetscFree(idxhit);CHKERRQ(ierr);
1543   PetscFunctionReturn(0);
1544 }
1545 
1546 /* Needed for MatMatMult_SeqAIJ_SeqAIJ_Combined() */
1547 /* Append value to an array if the value is not present yet. A bitarray */
1548 /* was used to determine if there is already an entry at this position. */
1549 void appendToArray(PetscInt val, PetscInt *array, PetscInt *cnzi)
1550 {
1551   array[(*cnzi)++] = val;
1552 }
1553 
1554 /* This algorithm combines the symbolic and numeric phase of matrix-matrix multiplication. */
1555 PetscErrorCode MatMatMult_SeqAIJ_SeqAIJ_Combined(Mat A,Mat B,PetscReal fill,Mat *C)
1556 {
1557   PetscErrorCode     ierr;
1558   PetscLogDouble     flops=0.0;
1559   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data, *b = (Mat_SeqAIJ*)B->data, *c;
1560   const PetscInt     *ai = a->i,*bi = b->i, *aj = a->j;
1561   PetscInt           *ci,*cj,*cj_i;
1562   PetscScalar        *ca, *ca_i;
1563   PetscInt           c_maxmem = 0, a_maxrownnz = 0, a_rownnz, a_col;
1564   PetscInt           am = A->rmap->N, bn = B->cmap->N, bm = B->rmap->N;
1565   PetscInt           i, k, ndouble = 0;
1566   PetscReal          afill;
1567   PetscScalar        *c_row_val_dense;
1568   PetscBool          *c_row_idx_flags;
1569   PetscInt           *aj_i = a->j;
1570   PetscScalar        *aa_i = a->a;
1571 
1572   PetscFunctionBegin;
1573   /* Step 1: Determine upper bounds on memory for C */
1574   for (i=0; i<am; i++) { /* iterate over all rows of A */
1575     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 */
1576     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
1577     a_rownnz = 0;
1578     for (k=0;k<anzi;++k) a_rownnz += bi[acol[k]+1] - bi[acol[k]];
1579     a_maxrownnz = PetscMax(a_maxrownnz, a_rownnz);
1580     c_maxmem += a_rownnz;
1581   }
1582   ierr = PetscMalloc1(am+1, &ci);               CHKERRQ(ierr);
1583   ierr = PetscMalloc1(bn, &c_row_val_dense);    CHKERRQ(ierr);
1584   ierr = PetscMalloc1(bn, &c_row_idx_flags);    CHKERRQ(ierr);
1585   ierr = PetscMalloc1(c_maxmem,&cj);            CHKERRQ(ierr);
1586   ierr = PetscMalloc1(c_maxmem,&ca);            CHKERRQ(ierr);
1587   ca_i = ca;
1588   cj_i = cj;
1589   ci[0] = 0;
1590   ierr = PetscMemzero(c_row_val_dense, bn * sizeof(PetscScalar));CHKERRQ(ierr);
1591   ierr = PetscMemzero(c_row_idx_flags, bn * sizeof(PetscBool));CHKERRQ(ierr);
1592   for (i=0; i<am; i++) {
1593     /* Step 2: Initialize the dense row vector for C  */
1594     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 */
1595     PetscInt cnzi           = 0;
1596     PetscInt *bj_i;
1597     PetscScalar *ba_i;
1598 
1599     /* Step 3: Do the numerical calculations */
1600     for (a_col=0; a_col<anzi; a_col++) {          /* iterate over all non zero values in a row of A */
1601       PetscInt a_col_index = aj_i[a_col];
1602       const PetscInt bnzi = bi[a_col_index+1] - bi[a_col_index];
1603       flops += 2*bnzi;
1604       bj_i = b->j + bi[a_col_index];   /* points to the current row in bj */
1605       ba_i = b->a + bi[a_col_index];   /* points to the current row in ba */
1606       for (k=0; k<bnzi; ++k) { /* iterate over all non zeros of this row in B */
1607         if (c_row_idx_flags[ bj_i[k] ] == PETSC_FALSE) {
1608           appendToArray(bj_i[k], cj_i, &cnzi);
1609           c_row_idx_flags[ bj_i[k] ] = PETSC_TRUE;
1610         }
1611         c_row_val_dense[ bj_i[k] ] += aa_i[a_col] * ba_i[k];
1612       }
1613     }
1614 
1615     /* Sort array */
1616     ierr = PetscSortInt(cnzi, cj_i);CHKERRQ(ierr);
1617     /* Step 4 */
1618     for (k=0; k < cnzi; k++) {
1619       ca_i[k] = c_row_val_dense[cj_i[k]];
1620       c_row_val_dense[cj_i[k]] = 0.;
1621       c_row_idx_flags[cj_i[k]] = PETSC_FALSE;
1622     }
1623     /* terminate current row */
1624     aa_i += anzi;
1625     aj_i += anzi;
1626     ca_i += cnzi;
1627     cj_i += cnzi;
1628     ci[i+1] = ci[i] + cnzi;
1629     flops += cnzi;
1630   }
1631 
1632   /* Step 5 */
1633   /* Create the new matrix */
1634   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
1635   ierr = MatSetBlockSizesFromMats(*C,A,B);CHKERRQ(ierr);
1636   ierr = MatSetType(*C,((PetscObject)A)->type_name);CHKERRQ(ierr);
1637 
1638   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
1639   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
1640   c          = (Mat_SeqAIJ*)((*C)->data);
1641   c->a       = ca;
1642   c->free_a  = PETSC_TRUE;
1643   c->free_ij = PETSC_TRUE;
1644   c->nonew   = 0;
1645 
1646   /* set MatInfo */
1647   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
1648   if (afill < 1.0) afill = 1.0;
1649   c->maxnz                     = ci[am];
1650   c->nz                        = ci[am];
1651   (*C)->info.mallocs           = ndouble;
1652   (*C)->info.fill_ratio_given  = fill;
1653   (*C)->info.fill_ratio_needed = afill;
1654 
1655   ierr = PetscFree(c_row_val_dense);CHKERRQ(ierr);
1656   ierr = PetscFree(c_row_idx_flags);CHKERRQ(ierr);
1657 
1658   ierr = MatAssemblyBegin(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1659   ierr = MatAssemblyEnd(*C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1660   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
1661   PetscFunctionReturn(0);
1662 }
1663