xref: /petsc/src/mat/impls/aij/seq/matmatmult.c (revision 58cf0668211c407e4235afbf5d5a3e67be12cf42)
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 <../src/mat/utils/petscheap.h>
10 #include <petscbt.h>
11 #include <../src/mat/impls/dense/seq/dense.h>
12 
13 static PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(Mat,Mat,PetscReal,Mat*);
14 
15 #undef __FUNCT__
16 #define __FUNCT__ "MatMatMult_SeqAIJ_SeqAIJ"
17 PetscErrorCode MatMatMult_SeqAIJ_SeqAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
18 {
19   PetscErrorCode ierr;
20   PetscBool      scalable=PETSC_FALSE,scalable_fast=PETSC_FALSE,heap = PETSC_FALSE,btheap = PETSC_FALSE,llcondensed = PETSC_FALSE;
21 
22   PetscFunctionBegin;
23   if (scall == MAT_INITIAL_MATRIX) {
24     ierr = PetscObjectOptionsBegin((PetscObject)A);CHKERRQ(ierr);
25     ierr = PetscOptionsBool("-matmatmult_scalable","Use a scalable but slower C=A*B","",scalable,&scalable,NULL);CHKERRQ(ierr);
26     ierr = PetscOptionsBool("-matmatmult_scalable_fast","Use a scalable but slower C=A*B","",scalable_fast,&scalable_fast,NULL);CHKERRQ(ierr);
27     ierr = PetscOptionsBool("-matmatmult_heap","Use heap implementation of symbolic factorization C=A*B","",heap,&heap,NULL);CHKERRQ(ierr);
28     ierr = PetscOptionsBool("-matmatmult_btheap","Use btheap implementation of symbolic factorization C=A*B","",btheap,&btheap,NULL);CHKERRQ(ierr);
29     ierr = PetscOptionsBool("-matmatmult_llcondensed","Use LLCondensed to for symbolic C=A*B","",llcondensed,&llcondensed,NULL);CHKERRQ(ierr);
30     ierr = PetscOptionsEnd();CHKERRQ(ierr);
31     if (scalable_fast) {
32       ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable_fast(A,B,fill,C);CHKERRQ(ierr);
33     } else if (scalable) {
34       ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable(A,B,fill,C);CHKERRQ(ierr);
35     } else if (heap) {
36       ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_Heap(A,B,fill,C);CHKERRQ(ierr);
37     } else if (btheap) {
38       ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_BTHeap(A,B,fill,C);CHKERRQ(ierr);
39     } else if (llcondensed) {
40       ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(A,B,fill,C);CHKERRQ(ierr);
41     } else {
42       ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(A,B,fill,C);CHKERRQ(ierr);
43     }
44   }
45 
46   ierr = (*(*C)->ops->matmultnumeric)(A,B,*C);CHKERRQ(ierr);
47   PetscFunctionReturn(0);
48 }
49 
50 #undef __FUNCT__
51 #define __FUNCT__ "MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed"
52 static PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_LLCondensed(Mat A,Mat B,PetscReal fill,Mat *C)
53 {
54   PetscErrorCode     ierr;
55   Mat_SeqAIJ         *a =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
56   PetscInt           *ai=a->i,*bi=b->i,*ci,*cj;
57   PetscInt           am =A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
58   PetscReal          afill;
59   PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,nlnk_max,*lnk,ndouble=0;
60   PetscBT            lnkbt;
61   PetscFreeSpaceList free_space=NULL,current_space=NULL;
62 
63   PetscFunctionBegin;
64   /* Get ci and cj */
65   /*---------------*/
66   /* Allocate ci array, arrays for fill computation and */
67   /* free space for accumulating nonzero column info */
68   ierr  = PetscMalloc(((am+1)+1)*sizeof(PetscInt),&ci);CHKERRQ(ierr);
69   ci[0] = 0;
70 
71   /* create and initialize a linked list */
72   nlnk_max = a->rmax*b->rmax;
73   if (!nlnk_max || nlnk_max > bn) nlnk_max = bn;
74   ierr = PetscLLCondensedCreate(nlnk_max,bn,&lnk,&lnkbt);CHKERRQ(ierr);
75 
76   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
77   ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+bi[bm])),&free_space);CHKERRQ(ierr);
78 
79   current_space = free_space;
80 
81   /* Determine ci and cj */
82   for (i=0; i<am; i++) {
83     anzi = ai[i+1] - ai[i];
84     aj   = a->j + ai[i];
85     for (j=0; j<anzi; j++) {
86       brow = aj[j];
87       bnzj = bi[brow+1] - bi[brow];
88       bj   = b->j + bi[brow];
89       /* add non-zero cols of B into the sorted linked list lnk */
90       ierr = PetscLLCondensedAddSorted(bnzj,bj,lnk,lnkbt);CHKERRQ(ierr);
91     }
92     cnzi = lnk[0];
93 
94     /* If free space is not available, make more free space */
95     /* Double the amount of total space in the list */
96     if (current_space->local_remaining<cnzi) {
97       ierr = PetscFreeSpaceGet(cnzi+current_space->total_array_size,&current_space);CHKERRQ(ierr);
98       ndouble++;
99     }
100 
101     /* Copy data into free space, then initialize lnk */
102     ierr = PetscLLCondensedClean(bn,cnzi,current_space->array,lnk,lnkbt);CHKERRQ(ierr);
103 
104     current_space->array           += cnzi;
105     current_space->local_used      += cnzi;
106     current_space->local_remaining -= cnzi;
107 
108     ci[i+1] = ci[i] + cnzi;
109   }
110 
111   /* Column indices are in the list of free space */
112   /* Allocate space for cj, initialize cj, and */
113   /* destroy list of free space and other temporary array(s) */
114   ierr = PetscMalloc((ci[am]+1)*sizeof(PetscInt),&cj);CHKERRQ(ierr);
115   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
116   ierr = PetscLLCondensedDestroy(lnk,lnkbt);CHKERRQ(ierr);
117 
118   /* put together the new symbolic matrix */
119   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
120 
121   (*C)->rmap->bs = A->rmap->bs;
122   (*C)->cmap->bs = B->cmap->bs;
123 
124   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
125   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
126   c                         = (Mat_SeqAIJ*)((*C)->data);
127   c->free_a                 = PETSC_FALSE;
128   c->free_ij                = PETSC_TRUE;
129   c->nonew                  = 0;
130   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ; /* fast, needs non-scalable O(bn) array 'abdense' */
131 
132   /* set MatInfo */
133   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
134   if (afill < 1.0) afill = 1.0;
135   c->maxnz                     = ci[am];
136   c->nz                        = ci[am];
137   (*C)->info.mallocs           = ndouble;
138   (*C)->info.fill_ratio_given  = fill;
139   (*C)->info.fill_ratio_needed = afill;
140 
141 #if defined(PETSC_USE_INFO)
142   if (ci[am]) {
143     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %G needed %G.\n",ndouble,fill,afill);CHKERRQ(ierr);
144     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%G,&C) for best performance.;\n",afill);CHKERRQ(ierr);
145   } else {
146     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
147   }
148 #endif
149   PetscFunctionReturn(0);
150 }
151 
152 #undef __FUNCT__
153 #define __FUNCT__ "MatMatMultNumeric_SeqAIJ_SeqAIJ"
154 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
155 {
156   PetscErrorCode ierr;
157   PetscLogDouble flops=0.0;
158   Mat_SeqAIJ     *a   = (Mat_SeqAIJ*)A->data;
159   Mat_SeqAIJ     *b   = (Mat_SeqAIJ*)B->data;
160   Mat_SeqAIJ     *c   = (Mat_SeqAIJ*)C->data;
161   PetscInt       *ai  =a->i,*aj=a->j,*bi=b->i,*bj=b->j,*bjj,*ci=c->i,*cj=c->j;
162   PetscInt       am   =A->rmap->n,cm=C->rmap->n;
163   PetscInt       i,j,k,anzi,bnzi,cnzi,brow;
164   PetscScalar    *aa=a->a,*ba=b->a,*baj,*ca,valtmp;
165   PetscScalar    *ab_dense;
166 
167   PetscFunctionBegin;
168   /* printf("MatMatMultNumeric_SeqAIJ_SeqAIJ...ca %p\n",c->a); */
169   if (!c->a) { /* first call of MatMatMultNumeric_SeqAIJ_SeqAIJ, allocate ca and matmult_abdense */
170     ierr      = PetscMalloc((ci[cm]+1)*sizeof(MatScalar),&ca);CHKERRQ(ierr);
171     c->a      = ca;
172     c->free_a = PETSC_TRUE;
173 
174     ierr = PetscMalloc(B->cmap->N*sizeof(PetscScalar),&ab_dense);CHKERRQ(ierr);
175     ierr = PetscMemzero(ab_dense,B->cmap->N*sizeof(PetscScalar));CHKERRQ(ierr);
176 
177     c->matmult_abdense = ab_dense;
178   } else {
179     ca       = c->a;
180     ab_dense = c->matmult_abdense;
181   }
182 
183   /* clean old values in C */
184   ierr = PetscMemzero(ca,ci[cm]*sizeof(MatScalar));CHKERRQ(ierr);
185   /* Traverse A row-wise. */
186   /* Build the ith row in C by summing over nonzero columns in A, */
187   /* the rows of B corresponding to nonzeros of A. */
188   for (i=0; i<am; i++) {
189     anzi = ai[i+1] - ai[i];
190     for (j=0; j<anzi; j++) {
191       brow = aj[j];
192       bnzi = bi[brow+1] - bi[brow];
193       bjj  = bj + bi[brow];
194       baj  = ba + bi[brow];
195       /* perform dense axpy */
196       valtmp = aa[j];
197       for (k=0; k<bnzi; k++) {
198         ab_dense[bjj[k]] += valtmp*baj[k];
199       }
200       flops += 2*bnzi;
201     }
202     aj += anzi; aa += anzi;
203 
204     cnzi = ci[i+1] - ci[i];
205     for (k=0; k<cnzi; k++) {
206       ca[k]          += ab_dense[cj[k]];
207       ab_dense[cj[k]] = 0.0; /* zero ab_dense */
208     }
209     flops += cnzi;
210     cj    += cnzi; ca += cnzi;
211   }
212   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
213   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
214   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
215   PetscFunctionReturn(0);
216 }
217 
218 #undef __FUNCT__
219 #define __FUNCT__ "MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable"
220 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable(Mat A,Mat B,Mat C)
221 {
222   PetscErrorCode ierr;
223   PetscLogDouble flops=0.0;
224   Mat_SeqAIJ     *a   = (Mat_SeqAIJ*)A->data;
225   Mat_SeqAIJ     *b   = (Mat_SeqAIJ*)B->data;
226   Mat_SeqAIJ     *c   = (Mat_SeqAIJ*)C->data;
227   PetscInt       *ai  = a->i,*aj=a->j,*bi=b->i,*bj=b->j,*bjj,*ci=c->i,*cj=c->j;
228   PetscInt       am   = A->rmap->N,cm=C->rmap->N;
229   PetscInt       i,j,k,anzi,bnzi,cnzi,brow;
230   PetscScalar    *aa=a->a,*ba=b->a,*baj,*ca=c->a,valtmp;
231   PetscInt       nextb;
232 
233   PetscFunctionBegin;
234   /* clean old values in C */
235   ierr = PetscMemzero(ca,ci[cm]*sizeof(MatScalar));CHKERRQ(ierr);
236   /* Traverse A row-wise. */
237   /* Build the ith row in C by summing over nonzero columns in A, */
238   /* the rows of B corresponding to nonzeros of A. */
239   for (i=0; i<am; i++) {
240     anzi = ai[i+1] - ai[i];
241     cnzi = ci[i+1] - ci[i];
242     for (j=0; j<anzi; j++) {
243       brow = aj[j];
244       bnzi = bi[brow+1] - bi[brow];
245       bjj  = bj + bi[brow];
246       baj  = ba + bi[brow];
247       /* perform sparse axpy */
248       valtmp = aa[j];
249       nextb  = 0;
250       for (k=0; nextb<bnzi; k++) {
251         if (cj[k] == bjj[nextb]) { /* ccol == bcol */
252           ca[k] += valtmp*baj[nextb++];
253         }
254       }
255       flops += 2*bnzi;
256     }
257     aj += anzi; aa += anzi;
258     cj += cnzi; ca += cnzi;
259   }
260 
261   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
262   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
263   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
264   PetscFunctionReturn(0);
265 }
266 
267 #undef __FUNCT__
268 #define __FUNCT__ "MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable_fast"
269 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable_fast(Mat A,Mat B,PetscReal fill,Mat *C)
270 {
271   PetscErrorCode     ierr;
272   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
273   PetscInt           *ai = a->i,*bi=b->i,*ci,*cj;
274   PetscInt           am  = A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
275   MatScalar          *ca;
276   PetscReal          afill;
277   PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,nlnk_max,*lnk,ndouble=0;
278   PetscFreeSpaceList free_space=NULL,current_space=NULL;
279 
280   PetscFunctionBegin;
281   /* Get ci and cj - same as MatMatMultSymbolic_SeqAIJ_SeqAIJ except using PetscLLxxx_fast() */
282   /*-----------------------------------------------------------------------------------------*/
283   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
284   ierr  = PetscMalloc(((am+1)+1)*sizeof(PetscInt),&ci);CHKERRQ(ierr);
285   ci[0] = 0;
286 
287   /* create and initialize a linked list */
288   nlnk_max = a->rmax*b->rmax;
289   if (!nlnk_max || nlnk_max > bn) nlnk_max = bn; /* in case rmax is not defined for A or B */
290   ierr = PetscLLCondensedCreate_fast(nlnk_max,&lnk);CHKERRQ(ierr);
291 
292   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
293   ierr          = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+bi[bm])),&free_space);CHKERRQ(ierr);
294   current_space = free_space;
295 
296   /* Determine ci and cj */
297   for (i=0; i<am; i++) {
298     anzi = ai[i+1] - ai[i];
299     aj   = a->j + ai[i];
300     for (j=0; j<anzi; j++) {
301       brow = aj[j];
302       bnzj = bi[brow+1] - bi[brow];
303       bj   = b->j + bi[brow];
304       /* add non-zero cols of B into the sorted linked list lnk */
305       ierr = PetscLLCondensedAddSorted_fast(bnzj,bj,lnk);CHKERRQ(ierr);
306     }
307     cnzi = lnk[1];
308 
309     /* If free space is not available, make more free space */
310     /* Double the amount of total space in the list */
311     if (current_space->local_remaining<cnzi) {
312       ierr = PetscFreeSpaceGet(cnzi+current_space->total_array_size,&current_space);CHKERRQ(ierr);
313       ndouble++;
314     }
315 
316     /* Copy data into free space, then initialize lnk */
317     ierr = PetscLLCondensedClean_fast(cnzi,current_space->array,lnk);CHKERRQ(ierr);
318 
319     current_space->array           += cnzi;
320     current_space->local_used      += cnzi;
321     current_space->local_remaining -= cnzi;
322 
323     ci[i+1] = ci[i] + cnzi;
324   }
325 
326   /* Column indices are in the list of free space */
327   /* Allocate space for cj, initialize cj, and */
328   /* destroy list of free space and other temporary array(s) */
329   ierr = PetscMalloc((ci[am]+1)*sizeof(PetscInt),&cj);CHKERRQ(ierr);
330   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
331   ierr = PetscLLCondensedDestroy_fast(lnk);CHKERRQ(ierr);
332 
333   /* Allocate space for ca */
334   ierr = PetscMalloc((ci[am]+1)*sizeof(MatScalar),&ca);CHKERRQ(ierr);
335   ierr = PetscMemzero(ca,(ci[am]+1)*sizeof(MatScalar));CHKERRQ(ierr);
336 
337   /* put together the new symbolic matrix */
338   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,ca,C);CHKERRQ(ierr);
339 
340   (*C)->rmap->bs = A->rmap->bs;
341   (*C)->cmap->bs = B->cmap->bs;
342 
343   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
344   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
345   c          = (Mat_SeqAIJ*)((*C)->data);
346   c->free_a  = PETSC_TRUE;
347   c->free_ij = PETSC_TRUE;
348   c->nonew   = 0;
349 
350   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable; /* slower, less memory */
351 
352   /* set MatInfo */
353   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
354   if (afill < 1.0) afill = 1.0;
355   c->maxnz                     = ci[am];
356   c->nz                        = ci[am];
357   (*C)->info.mallocs           = ndouble;
358   (*C)->info.fill_ratio_given  = fill;
359   (*C)->info.fill_ratio_needed = afill;
360 
361 #if defined(PETSC_USE_INFO)
362   if (ci[am]) {
363     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %G needed %G.\n",ndouble,fill,afill);CHKERRQ(ierr);
364     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%G,&C) for best performance.;\n",afill);CHKERRQ(ierr);
365   } else {
366     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
367   }
368 #endif
369   PetscFunctionReturn(0);
370 }
371 
372 
373 #undef __FUNCT__
374 #define __FUNCT__ "MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable"
375 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Scalable(Mat A,Mat B,PetscReal fill,Mat *C)
376 {
377   PetscErrorCode     ierr;
378   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
379   PetscInt           *ai = a->i,*bi=b->i,*ci,*cj;
380   PetscInt           am  = A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
381   MatScalar          *ca;
382   PetscReal          afill;
383   PetscInt           i,j,anzi,brow,bnzj,cnzi,*bj,*aj,nlnk_max,*lnk,ndouble=0;
384   PetscFreeSpaceList free_space=NULL,current_space=NULL;
385 
386   PetscFunctionBegin;
387   /* Get ci and cj - same as MatMatMultSymbolic_SeqAIJ_SeqAIJ except using PetscLLxxx_Scalalbe() */
388   /*---------------------------------------------------------------------------------------------*/
389   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
390   ierr  = PetscMalloc(((am+1)+1)*sizeof(PetscInt),&ci);CHKERRQ(ierr);
391   ci[0] = 0;
392 
393   /* create and initialize a linked list */
394   nlnk_max = a->rmax*b->rmax;
395   if (!nlnk_max || nlnk_max > bn) nlnk_max = bn; /* in case rmax is not defined for A or B */
396   ierr = PetscLLCondensedCreate_Scalable(nlnk_max,&lnk);CHKERRQ(ierr);
397 
398   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
399   ierr          = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+bi[bm])),&free_space);CHKERRQ(ierr);
400   current_space = free_space;
401 
402   /* Determine ci and cj */
403   for (i=0; i<am; i++) {
404     anzi = ai[i+1] - ai[i];
405     aj   = a->j + ai[i];
406     for (j=0; j<anzi; j++) {
407       brow = aj[j];
408       bnzj = bi[brow+1] - bi[brow];
409       bj   = b->j + bi[brow];
410       /* add non-zero cols of B into the sorted linked list lnk */
411       ierr = PetscLLCondensedAddSorted_Scalable(bnzj,bj,lnk);CHKERRQ(ierr);
412     }
413     cnzi = lnk[0];
414 
415     /* If free space is not available, make more free space */
416     /* Double the amount of total space in the list */
417     if (current_space->local_remaining<cnzi) {
418       ierr = PetscFreeSpaceGet(cnzi+current_space->total_array_size,&current_space);CHKERRQ(ierr);
419       ndouble++;
420     }
421 
422     /* Copy data into free space, then initialize lnk */
423     ierr = PetscLLCondensedClean_Scalable(cnzi,current_space->array,lnk);CHKERRQ(ierr);
424 
425     current_space->array           += cnzi;
426     current_space->local_used      += cnzi;
427     current_space->local_remaining -= cnzi;
428 
429     ci[i+1] = ci[i] + cnzi;
430   }
431 
432   /* Column indices are in the list of free space */
433   /* Allocate space for cj, initialize cj, and */
434   /* destroy list of free space and other temporary array(s) */
435   ierr = PetscMalloc((ci[am]+1)*sizeof(PetscInt),&cj);CHKERRQ(ierr);
436   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
437   ierr = PetscLLCondensedDestroy_Scalable(lnk);CHKERRQ(ierr);
438 
439   /* Allocate space for ca */
440   /*-----------------------*/
441   ierr = PetscMalloc((ci[am]+1)*sizeof(MatScalar),&ca);CHKERRQ(ierr);
442   ierr = PetscMemzero(ca,(ci[am]+1)*sizeof(MatScalar));CHKERRQ(ierr);
443 
444   /* put together the new symbolic matrix */
445   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,ca,C);CHKERRQ(ierr);
446 
447   (*C)->rmap->bs = A->rmap->bs;
448   (*C)->cmap->bs = B->cmap->bs;
449 
450   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
451   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
452   c          = (Mat_SeqAIJ*)((*C)->data);
453   c->free_a  = PETSC_TRUE;
454   c->free_ij = PETSC_TRUE;
455   c->nonew   = 0;
456 
457   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ_Scalable; /* slower, less memory */
458 
459   /* set MatInfo */
460   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
461   if (afill < 1.0) afill = 1.0;
462   c->maxnz                     = ci[am];
463   c->nz                        = ci[am];
464   (*C)->info.mallocs           = ndouble;
465   (*C)->info.fill_ratio_given  = fill;
466   (*C)->info.fill_ratio_needed = afill;
467 
468 #if defined(PETSC_USE_INFO)
469   if (ci[am]) {
470     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %G needed %G.\n",ndouble,fill,afill);CHKERRQ(ierr);
471     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%G,&C) for best performance.;\n",afill);CHKERRQ(ierr);
472   } else {
473     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
474   }
475 #endif
476   PetscFunctionReturn(0);
477 }
478 
479 #undef __FUNCT__
480 #define __FUNCT__ "MatMatMultSymbolic_SeqAIJ_SeqAIJ_Heap"
481 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_Heap(Mat A,Mat B,PetscReal fill,Mat *C)
482 {
483   PetscErrorCode     ierr;
484   Mat_SeqAIJ         *a = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
485   const PetscInt     *ai=a->i,*bi=b->i,*aj=a->j,*bj=b->j;
486   PetscInt           *ci,*cj,*bb;
487   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
488   PetscReal          afill;
489   PetscInt           i,j,col,ndouble = 0;
490   PetscFreeSpaceList free_space=NULL,current_space=NULL;
491   PetscHeap          h;
492 
493   PetscFunctionBegin;
494   /* Get ci and cj - by merging sorted rows using a heap */
495   /*---------------------------------------------------------------------------------------------*/
496   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
497   ierr  = PetscMalloc(((am+1)+1)*sizeof(PetscInt),&ci);CHKERRQ(ierr);
498   ci[0] = 0;
499 
500   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
501   ierr          = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+bi[bm])),&free_space);CHKERRQ(ierr);
502   current_space = free_space;
503 
504   ierr = PetscHeapCreate(a->rmax,&h);CHKERRQ(ierr);
505   ierr = PetscMalloc(a->rmax*sizeof(PetscInt),&bb);CHKERRQ(ierr);
506 
507   /* Determine ci and cj */
508   for (i=0; i<am; i++) {
509     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 */
510     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
511     ci[i+1] = ci[i];
512     /* Populate the min heap */
513     for (j=0; j<anzi; j++) {
514       bb[j] = bi[acol[j]];         /* bb points at the start of the row */
515       if (bb[j] < bi[acol[j]+1]) { /* Add if row is nonempty */
516         ierr = PetscHeapAdd(h,j,bj[bb[j]++]);CHKERRQ(ierr);
517       }
518     }
519     /* Pick off the min element, adding it to free space */
520     ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
521     while (j >= 0) {
522       if (current_space->local_remaining < 1) { /* double the size, but don't exceed 16 MiB */
523         ierr = PetscFreeSpaceGet(PetscMin(2*current_space->total_array_size,16 << 20),&current_space);CHKERRQ(ierr);
524         ndouble++;
525       }
526       *(current_space->array++) = col;
527       current_space->local_used++;
528       current_space->local_remaining--;
529       ci[i+1]++;
530 
531       /* stash if anything else remains in this row of B */
532       if (bb[j] < bi[acol[j]+1]) {ierr = PetscHeapStash(h,j,bj[bb[j]++]);CHKERRQ(ierr);}
533       while (1) {               /* pop and stash any other rows of B that also had an entry in this column */
534         PetscInt j2,col2;
535         ierr = PetscHeapPeek(h,&j2,&col2);CHKERRQ(ierr);
536         if (col2 != col) break;
537         ierr = PetscHeapPop(h,&j2,&col2);CHKERRQ(ierr);
538         if (bb[j2] < bi[acol[j2]+1]) {ierr = PetscHeapStash(h,j2,bj[bb[j2]++]);CHKERRQ(ierr);}
539       }
540       /* Put any stashed elements back into the min heap */
541       ierr = PetscHeapUnstash(h);CHKERRQ(ierr);
542       ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
543     }
544   }
545   ierr = PetscFree(bb);CHKERRQ(ierr);
546   ierr = PetscHeapDestroy(&h);CHKERRQ(ierr);
547 
548   /* Column indices are in the list of free space */
549   /* Allocate space for cj, initialize cj, and */
550   /* destroy list of free space and other temporary array(s) */
551   ierr = PetscMalloc(ci[am]*sizeof(PetscInt),&cj);CHKERRQ(ierr);
552   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
553 
554   /* put together the new symbolic matrix */
555   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
556 
557   (*C)->rmap->bs = A->rmap->bs;
558   (*C)->cmap->bs = B->cmap->bs;
559 
560   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
561   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
562   c          = (Mat_SeqAIJ*)((*C)->data);
563   c->free_a  = PETSC_TRUE;
564   c->free_ij = PETSC_TRUE;
565   c->nonew   = 0;
566 
567   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ;
568 
569   /* set MatInfo */
570   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
571   if (afill < 1.0) afill = 1.0;
572   c->maxnz                     = ci[am];
573   c->nz                        = ci[am];
574   (*C)->info.mallocs           = ndouble;
575   (*C)->info.fill_ratio_given  = fill;
576   (*C)->info.fill_ratio_needed = afill;
577 
578 #if defined(PETSC_USE_INFO)
579   if (ci[am]) {
580     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %G needed %G.\n",ndouble,fill,afill);CHKERRQ(ierr);
581     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%G,&C) for best performance.;\n",afill);CHKERRQ(ierr);
582   } else {
583     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
584   }
585 #endif
586   PetscFunctionReturn(0);
587 }
588 
589 #undef __FUNCT__
590 #define __FUNCT__ "MatMatMultSymbolic_SeqAIJ_SeqAIJ_BTHeap"
591 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ_BTHeap(Mat A,Mat B,PetscReal fill,Mat *C)
592 {
593   PetscErrorCode     ierr;
594   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
595   const PetscInt     *ai = a->i,*bi=b->i,*aj=a->j,*bj=b->j;
596   PetscInt           *ci,*cj,*bb;
597   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
598   PetscReal          afill;
599   PetscInt           i,j,col,ndouble = 0;
600   PetscFreeSpaceList free_space=NULL,current_space=NULL;
601   PetscHeap          h;
602   PetscBT            bt;
603 
604   PetscFunctionBegin;
605   /* Get ci and cj - using a heap for the sorted rows, but use BT so that each index is only added once */
606   /*---------------------------------------------------------------------------------------------*/
607   /* Allocate arrays for fill computation and free space for accumulating nonzero column */
608   ierr  = PetscMalloc(((am+1)+1)*sizeof(PetscInt),&ci);CHKERRQ(ierr);
609   ci[0] = 0;
610 
611   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
612   ierr = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+bi[bm])),&free_space);CHKERRQ(ierr);
613 
614   current_space = free_space;
615 
616   ierr = PetscHeapCreate(a->rmax,&h);CHKERRQ(ierr);
617   ierr = PetscMalloc(a->rmax*sizeof(PetscInt),&bb);CHKERRQ(ierr);
618   ierr = PetscBTCreate(bn,&bt);CHKERRQ(ierr);
619 
620   /* Determine ci and cj */
621   for (i=0; i<am; i++) {
622     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 */
623     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
624     const PetscInt *fptr = current_space->array; /* Save beginning of the row so we can clear the BT later */
625     ci[i+1] = ci[i];
626     /* Populate the min heap */
627     for (j=0; j<anzi; j++) {
628       PetscInt brow = acol[j];
629       for (bb[j] = bi[brow]; bb[j] < bi[brow+1]; bb[j]++) {
630         PetscInt bcol = bj[bb[j]];
631         if (!PetscBTLookupSet(bt,bcol)) { /* new entry */
632           ierr = PetscHeapAdd(h,j,bcol);CHKERRQ(ierr);
633           bb[j]++;
634           break;
635         }
636       }
637     }
638     /* Pick off the min element, adding it to free space */
639     ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
640     while (j >= 0) {
641       if (current_space->local_remaining < 1) { /* double the size, but don't exceed 16 MiB */
642         fptr = NULL;                      /* need PetscBTMemzero */
643         ierr = PetscFreeSpaceGet(PetscMin(2*current_space->total_array_size,16 << 20),&current_space);CHKERRQ(ierr);
644         ndouble++;
645       }
646       *(current_space->array++) = col;
647       current_space->local_used++;
648       current_space->local_remaining--;
649       ci[i+1]++;
650 
651       /* stash if anything else remains in this row of B */
652       for (; bb[j] < bi[acol[j]+1]; bb[j]++) {
653         PetscInt bcol = bj[bb[j]];
654         if (!PetscBTLookupSet(bt,bcol)) { /* new entry */
655           ierr = PetscHeapAdd(h,j,bcol);CHKERRQ(ierr);
656           bb[j]++;
657           break;
658         }
659       }
660       ierr = PetscHeapPop(h,&j,&col);CHKERRQ(ierr);
661     }
662     if (fptr) {                 /* Clear the bits for this row */
663       for (; fptr<current_space->array; fptr++) {ierr = PetscBTClear(bt,*fptr);CHKERRQ(ierr);}
664     } else {                    /* We reallocated so we don't remember (easily) how to clear only the bits we changed */
665       ierr = PetscBTMemzero(bn,bt);CHKERRQ(ierr);
666     }
667   }
668   ierr = PetscFree(bb);CHKERRQ(ierr);
669   ierr = PetscHeapDestroy(&h);CHKERRQ(ierr);
670   ierr = PetscBTDestroy(&bt);CHKERRQ(ierr);
671 
672   /* Column indices are in the list of free space */
673   /* Allocate space for cj, initialize cj, and */
674   /* destroy list of free space and other temporary array(s) */
675   ierr = PetscMalloc(ci[am]*sizeof(PetscInt),&cj);CHKERRQ(ierr);
676   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
677 
678   /* put together the new symbolic matrix */
679   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
680 
681   (*C)->rmap->bs = A->rmap->bs;
682   (*C)->cmap->bs = B->cmap->bs;
683 
684   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
685   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
686   c          = (Mat_SeqAIJ*)((*C)->data);
687   c->free_a  = PETSC_TRUE;
688   c->free_ij = PETSC_TRUE;
689   c->nonew   = 0;
690 
691   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ;
692 
693   /* set MatInfo */
694   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
695   if (afill < 1.0) afill = 1.0;
696   c->maxnz                     = ci[am];
697   c->nz                        = ci[am];
698   (*C)->info.mallocs           = ndouble;
699   (*C)->info.fill_ratio_given  = fill;
700   (*C)->info.fill_ratio_needed = afill;
701 
702 #if defined(PETSC_USE_INFO)
703   if (ci[am]) {
704     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %G needed %G.\n",ndouble,fill,afill);CHKERRQ(ierr);
705     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%G,&C) for best performance.;\n",afill);CHKERRQ(ierr);
706   } else {
707     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
708   }
709 #endif
710   PetscFunctionReturn(0);
711 }
712 
713 #undef __FUNCT__
714 #define __FUNCT__ "MatMatMultSymbolic_SeqAIJ_SeqAIJ"
715 /* concatenate unique entries and then sort */
716 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat *C)
717 {
718   PetscErrorCode     ierr;
719   Mat_SeqAIJ         *a  = (Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
720   const PetscInt     *ai = a->i,*bi=b->i,*aj=a->j,*bj=b->j;
721   PetscInt           *ci,*cj;
722   PetscInt           am=A->rmap->N,bn=B->cmap->N,bm=B->rmap->N;
723   PetscReal          afill;
724   PetscInt           i,j,ndouble = 0;
725   PetscSegBuffer     seg,segrow;
726   char               *seen;
727 
728   PetscFunctionBegin;
729   ierr  = PetscMalloc((am+1)*sizeof(PetscInt),&ci);CHKERRQ(ierr);
730   ci[0] = 0;
731 
732   /* Initial FreeSpace size is fill*(nnz(A)+nnz(B)) */
733   ierr = PetscSegBufferCreate(sizeof(PetscInt),(PetscInt)(fill*(ai[am]+bi[bm])),&seg);CHKERRQ(ierr);
734   ierr = PetscSegBufferCreate(sizeof(PetscInt),100,&segrow);CHKERRQ(ierr);
735   ierr = PetscMalloc(bn*sizeof(char),&seen);CHKERRQ(ierr);
736   ierr = PetscMemzero(seen,bn*sizeof(char));CHKERRQ(ierr);
737 
738   /* Determine ci and cj */
739   for (i=0; i<am; i++) {
740     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 */
741     const PetscInt *acol = aj + ai[i]; /* column indices of nonzero entries in this row */
742     PetscInt packlen = 0,*PETSC_RESTRICT crow;
743     /* Pack segrow */
744     for (j=0; j<anzi; j++) {
745       PetscInt brow = acol[j],bjstart = bi[brow],bjend = bi[brow+1],k;
746       for (k=bjstart; k<bjend; k++) {
747         PetscInt bcol = bj[k];
748         if (!seen[bcol]) { /* new entry */
749           PetscInt *PETSC_RESTRICT slot;
750           ierr = PetscSegBufferGetInts(segrow,1,&slot);CHKERRQ(ierr);
751           *slot = bcol;
752           seen[bcol] = 1;
753           packlen++;
754         }
755       }
756     }
757     ierr = PetscSegBufferGetInts(seg,packlen,&crow);CHKERRQ(ierr);
758     ierr = PetscSegBufferExtractTo(segrow,crow);CHKERRQ(ierr);
759     ierr = PetscSortInt(packlen,crow);CHKERRQ(ierr);
760     ci[i+1] = ci[i] + packlen;
761     for (j=0; j<packlen; j++) seen[crow[j]] = 0;
762   }
763   ierr = PetscSegBufferDestroy(&segrow);CHKERRQ(ierr);
764   ierr = PetscFree(seen);CHKERRQ(ierr);
765 
766   /* Column indices are in the segmented buffer */
767   ierr = PetscSegBufferExtractAlloc(seg,&cj);CHKERRQ(ierr);
768   ierr = PetscSegBufferDestroy(&seg);CHKERRQ(ierr);
769 
770   /* put together the new symbolic matrix */
771   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bn,ci,cj,NULL,C);CHKERRQ(ierr);
772 
773   (*C)->rmap->bs = A->rmap->bs;
774   (*C)->cmap->bs = B->cmap->bs;
775 
776   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
777   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
778   c          = (Mat_SeqAIJ*)((*C)->data);
779   c->free_a  = PETSC_TRUE;
780   c->free_ij = PETSC_TRUE;
781   c->nonew   = 0;
782 
783   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqAIJ;
784 
785   /* set MatInfo */
786   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
787   if (afill < 1.0) afill = 1.0;
788   c->maxnz                     = ci[am];
789   c->nz                        = ci[am];
790   (*C)->info.mallocs           = ndouble;
791   (*C)->info.fill_ratio_given  = fill;
792   (*C)->info.fill_ratio_needed = afill;
793 
794 #if defined(PETSC_USE_INFO)
795   if (ci[am]) {
796     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %G needed %G.\n",ndouble,fill,afill);CHKERRQ(ierr);
797     ierr = PetscInfo1((*C),"Use MatMatMult(A,B,MatReuse,%G,&C) for best performance.;\n",afill);CHKERRQ(ierr);
798   } else {
799     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
800   }
801 #endif
802   PetscFunctionReturn(0);
803 }
804 
805 /* This routine is not used. Should be removed! */
806 #undef __FUNCT__
807 #define __FUNCT__ "MatMatTransposeMult_SeqAIJ_SeqAIJ"
808 PetscErrorCode MatMatTransposeMult_SeqAIJ_SeqAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
809 {
810   PetscErrorCode ierr;
811 
812   PetscFunctionBegin;
813   if (scall == MAT_INITIAL_MATRIX) {
814     ierr = MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ(A,B,fill,C);CHKERRQ(ierr);
815   }
816   ierr = MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ(A,B,*C);CHKERRQ(ierr);
817   PetscFunctionReturn(0);
818 }
819 
820 #undef __FUNCT__
821 #define __FUNCT__ "PetscContainerDestroy_Mat_MatMatTransMult"
822 PetscErrorCode PetscContainerDestroy_Mat_MatMatTransMult(void *ptr)
823 {
824   PetscErrorCode      ierr;
825   Mat_MatMatTransMult *multtrans=(Mat_MatMatTransMult*)ptr;
826 
827   PetscFunctionBegin;
828   ierr = MatTransposeColoringDestroy(&multtrans->matcoloring);CHKERRQ(ierr);
829   ierr = MatDestroy(&multtrans->Bt_den);CHKERRQ(ierr);
830   ierr = MatDestroy(&multtrans->ABt_den);CHKERRQ(ierr);
831   ierr = PetscFree(multtrans);CHKERRQ(ierr);
832   PetscFunctionReturn(0);
833 }
834 
835 #undef __FUNCT__
836 #define __FUNCT__ "MatDestroy_SeqAIJ_MatMatMultTrans"
837 PetscErrorCode MatDestroy_SeqAIJ_MatMatMultTrans(Mat A)
838 {
839   PetscErrorCode      ierr;
840   PetscContainer      container;
841   Mat_MatMatTransMult *multtrans=NULL;
842 
843   PetscFunctionBegin;
844   ierr = PetscObjectQuery((PetscObject)A,"Mat_MatMatTransMult",(PetscObject*)&container);CHKERRQ(ierr);
845   if (!container) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Container does not exit");
846   ierr = PetscContainerGetPointer(container,(void**)&multtrans);CHKERRQ(ierr);
847 
848   A->ops->destroy = multtrans->destroy;
849   if (A->ops->destroy) {
850     ierr = (*A->ops->destroy)(A);CHKERRQ(ierr);
851   }
852   ierr = PetscObjectCompose((PetscObject)A,"Mat_MatMatTransMult",0);CHKERRQ(ierr);
853   PetscFunctionReturn(0);
854 }
855 
856 #undef __FUNCT__
857 #define __FUNCT__ "MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ"
858 PetscErrorCode MatMatTransposeMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat *C)
859 {
860   PetscErrorCode      ierr;
861   Mat                 Bt;
862   PetscInt            *bti,*btj;
863   Mat_MatMatTransMult *multtrans;
864   PetscContainer      container;
865 
866   PetscFunctionBegin;
867   /* create symbolic Bt */
868   ierr = MatGetSymbolicTranspose_SeqAIJ(B,&bti,&btj);CHKERRQ(ierr);
869   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,B->cmap->n,B->rmap->n,bti,btj,NULL,&Bt);CHKERRQ(ierr);
870 
871   Bt->rmap->bs = A->cmap->bs;
872   Bt->cmap->bs = B->cmap->bs;
873 
874   /* get symbolic C=A*Bt */
875   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(A,Bt,fill,C);CHKERRQ(ierr);
876 
877   /* create a supporting struct for reuse intermidiate dense matrices with matcoloring */
878   ierr = PetscNew(Mat_MatMatTransMult,&multtrans);CHKERRQ(ierr);
879 
880   /* attach the supporting struct to C */
881   ierr = PetscContainerCreate(PETSC_COMM_SELF,&container);CHKERRQ(ierr);
882   ierr = PetscContainerSetPointer(container,multtrans);CHKERRQ(ierr);
883   ierr = PetscContainerSetUserDestroy(container,PetscContainerDestroy_Mat_MatMatTransMult);CHKERRQ(ierr);
884   ierr = PetscObjectCompose((PetscObject)(*C),"Mat_MatMatTransMult",(PetscObject)container);CHKERRQ(ierr);
885   ierr = PetscContainerDestroy(&container);CHKERRQ(ierr);
886 
887   multtrans->usecoloring = PETSC_FALSE;
888   multtrans->destroy     = (*C)->ops->destroy;
889   (*C)->ops->destroy     = MatDestroy_SeqAIJ_MatMatMultTrans;
890 
891   ierr = PetscOptionsGetBool(NULL,"-matmattransmult_color",&multtrans->usecoloring,NULL);CHKERRQ(ierr);
892   if (multtrans->usecoloring) {
893     /* Create MatTransposeColoring from symbolic C=A*B^T */
894     MatTransposeColoring matcoloring;
895     ISColoring           iscoloring;
896     Mat                  Bt_dense,C_dense;
897 
898     ierr = MatGetColoring(*C,MATCOLORINGLF,&iscoloring);CHKERRQ(ierr);
899     ierr = MatTransposeColoringCreate(*C,iscoloring,&matcoloring);CHKERRQ(ierr);
900 
901     multtrans->matcoloring = matcoloring;
902 
903     ierr = ISColoringDestroy(&iscoloring);CHKERRQ(ierr);
904 
905     /* Create Bt_dense and C_dense = A*Bt_dense */
906     ierr = MatCreate(PETSC_COMM_SELF,&Bt_dense);CHKERRQ(ierr);
907     ierr = MatSetSizes(Bt_dense,A->cmap->n,matcoloring->ncolors,A->cmap->n,matcoloring->ncolors);CHKERRQ(ierr);
908     ierr = MatSetType(Bt_dense,MATSEQDENSE);CHKERRQ(ierr);
909     ierr = MatSeqDenseSetPreallocation(Bt_dense,NULL);CHKERRQ(ierr);
910 
911     Bt_dense->assembled = PETSC_TRUE;
912     multtrans->Bt_den   = Bt_dense;
913 
914     ierr = MatCreate(PETSC_COMM_SELF,&C_dense);CHKERRQ(ierr);
915     ierr = MatSetSizes(C_dense,A->rmap->n,matcoloring->ncolors,A->rmap->n,matcoloring->ncolors);CHKERRQ(ierr);
916     ierr = MatSetType(C_dense,MATSEQDENSE);CHKERRQ(ierr);
917     ierr = MatSeqDenseSetPreallocation(C_dense,NULL);CHKERRQ(ierr);
918 
919     Bt_dense->assembled = PETSC_TRUE;
920     multtrans->ABt_den  = C_dense;
921 
922 #if defined(PETSC_USE_INFO)
923     {
924       Mat_SeqAIJ *c = (Mat_SeqAIJ*)(*C)->data;
925       ierr = PetscInfo5(*C,"Bt_dense: %D,%D; Cnz %D / (cm*ncolors %D) = %g\n",A->cmap->n,matcoloring->ncolors,c->nz,A->rmap->n*matcoloring->ncolors,(PetscReal)(c->nz)/(A->rmap->n*matcoloring->ncolors));CHKERRQ(ierr);
926     }
927 #endif
928   }
929   /* clean up */
930   ierr = MatDestroy(&Bt);CHKERRQ(ierr);
931   ierr = MatRestoreSymbolicTranspose_SeqAIJ(B,&bti,&btj);CHKERRQ(ierr);
932 
933 
934 
935 #if defined(INEFFICIENT_ALGORITHM)
936   /* The algorithm below computes am*bm sparse inner-product - inefficient! It will be deleted later. */
937   PetscFreeSpaceList free_space=NULL,current_space=NULL;
938   Mat_SeqAIJ         *a        =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c;
939   PetscInt           *ai       =a->i,*aj=a->j,*bi=b->i,*bj=b->j,*ci,*cj,*acol,*bcol;
940   PetscInt           am        =A->rmap->N,bm=B->rmap->N;
941   PetscInt           i,j,anzi,bnzj,cnzi,nlnk,*lnk,nspacedouble=0,ka,kb,index[1];
942   MatScalar          *ca;
943   PetscBT            lnkbt;
944   PetscReal          afill;
945 
946   /* Allocate row pointer array ci  */
947   ierr  = PetscMalloc(((am+1)+1)*sizeof(PetscInt),&ci);CHKERRQ(ierr);
948   ci[0] = 0;
949 
950   /* Create and initialize a linked list for C columns */
951   nlnk = bm+1;
952   ierr = PetscLLCreate(bm,bm,nlnk,lnk,lnkbt);CHKERRQ(ierr);
953 
954   /* Initial FreeSpace with size fill*(nnz(A)+nnz(B)) */
955   ierr          = PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+bi[bm])),&free_space);CHKERRQ(ierr);
956   current_space = free_space;
957 
958   /* Determine symbolic info for each row of the product A*B^T: */
959   for (i=0; i<am; i++) {
960     anzi = ai[i+1] - ai[i];
961     cnzi = 0;
962     acol = aj + ai[i];
963     for (j=0; j<bm; j++) {
964       bnzj = bi[j+1] - bi[j];
965       bcol = bj + bi[j];
966       /* sparse inner-product c(i,j)=A[i,:]*B[j,:]^T */
967       ka = 0; kb = 0;
968       while (ka < anzi && kb < bnzj) {
969         while (acol[ka] < bcol[kb] && ka < anzi) ka++;
970         if (ka == anzi) break;
971         while (acol[ka] > bcol[kb] && kb < bnzj) kb++;
972         if (kb == bnzj) break;
973         if (acol[ka] == bcol[kb]) { /* add nonzero c(i,j) to lnk */
974           index[0] = j;
975           ierr     = PetscLLAdd(1,index,bm,nlnk,lnk,lnkbt);CHKERRQ(ierr);
976           cnzi++;
977           break;
978         }
979       }
980     }
981 
982     /* If free space is not available, make more free space */
983     /* Double the amount of total space in the list */
984     if (current_space->local_remaining<cnzi) {
985       ierr = PetscFreeSpaceGet(cnzi+current_space->total_array_size,&current_space);CHKERRQ(ierr);
986       nspacedouble++;
987     }
988 
989     /* Copy data into free space, then initialize lnk */
990     ierr = PetscLLClean(bm,bm,cnzi,lnk,current_space->array,lnkbt);CHKERRQ(ierr);
991 
992     current_space->array           += cnzi;
993     current_space->local_used      += cnzi;
994     current_space->local_remaining -= cnzi;
995 
996     ci[i+1] = ci[i] + cnzi;
997   }
998 
999 
1000   /* Column indices are in the list of free space.
1001      Allocate array cj, copy column indices to cj, and destroy list of free space */
1002   ierr = PetscMalloc((ci[am]+1)*sizeof(PetscInt),&cj);CHKERRQ(ierr);
1003   ierr = PetscFreeSpaceContiguous(&free_space,cj);CHKERRQ(ierr);
1004   ierr = PetscLLDestroy(lnk,lnkbt);CHKERRQ(ierr);
1005 
1006   /* Allocate space for ca */
1007   ierr = PetscMalloc((ci[am]+1)*sizeof(MatScalar),&ca);CHKERRQ(ierr);
1008   ierr = PetscMemzero(ca,(ci[am]+1)*sizeof(MatScalar));CHKERRQ(ierr);
1009 
1010   /* put together the new symbolic matrix */
1011   ierr = MatCreateSeqAIJWithArrays(PetscObjectComm((PetscObject)A),am,bm,ci,cj,ca,C);CHKERRQ(ierr);
1012 
1013   (*C)->rmap->bs = A->cmap->bs;
1014   (*C)->cmap->bs = B->cmap->bs;
1015 
1016   /* MatCreateSeqAIJWithArrays flags matrix so PETSc doesn't free the user's arrays. */
1017   /* These are PETSc arrays, so change flags so arrays can be deleted by PETSc */
1018   c          = (Mat_SeqAIJ*)((*C)->data);
1019   c->free_a  = PETSC_TRUE;
1020   c->free_ij = PETSC_TRUE;
1021   c->nonew   = 0;
1022 
1023   /* set MatInfo */
1024   afill = (PetscReal)ci[am]/(ai[am]+bi[bm]) + 1.e-5;
1025   if (afill < 1.0) afill = 1.0;
1026   c->maxnz                     = ci[am];
1027   c->nz                        = ci[am];
1028   (*C)->info.mallocs           = nspacedouble;
1029   (*C)->info.fill_ratio_given  = fill;
1030   (*C)->info.fill_ratio_needed = afill;
1031 
1032 #if defined(PETSC_USE_INFO)
1033   if (ci[am]) {
1034     ierr = PetscInfo3((*C),"Reallocs %D; Fill ratio: given %G needed %G.\n",nspacedouble,fill,afill);CHKERRQ(ierr);
1035     ierr = PetscInfo1((*C),"Use MatMatTransposeMult(A,B,MatReuse,%G,&C) for best performance.;\n",afill);CHKERRQ(ierr);
1036   } else {
1037     ierr = PetscInfo((*C),"Empty matrix product\n");CHKERRQ(ierr);
1038   }
1039 #endif
1040 #endif
1041   PetscFunctionReturn(0);
1042 }
1043 
1044 /* #define USE_ARRAY - for sparse dot product. Slower than !USE_ARRAY */
1045 #undef __FUNCT__
1046 #define __FUNCT__ "MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ"
1047 PetscErrorCode MatMatTransposeMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
1048 {
1049   PetscErrorCode      ierr;
1050   Mat_SeqAIJ          *a   =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c=(Mat_SeqAIJ*)C->data;
1051   PetscInt            *ai  =a->i,*aj=a->j,*bi=b->i,*bj=b->j,anzi,bnzj,nexta,nextb,*acol,*bcol,brow;
1052   PetscInt            cm   =C->rmap->n,*ci=c->i,*cj=c->j,i,j,cnzi,*ccol;
1053   PetscLogDouble      flops=0.0;
1054   MatScalar           *aa  =a->a,*aval,*ba=b->a,*bval,*ca,*cval;
1055   Mat_MatMatTransMult *multtrans;
1056   PetscContainer      container;
1057 #if defined(USE_ARRAY)
1058   MatScalar *spdot;
1059 #endif
1060 
1061   PetscFunctionBegin;
1062   /* clear old values in C */
1063   if (!c->a) {
1064     ierr      = PetscMalloc((ci[cm]+1)*sizeof(MatScalar),&ca);CHKERRQ(ierr);
1065     c->a      = ca;
1066     c->free_a = PETSC_TRUE;
1067   } else {
1068     ca =  c->a;
1069   }
1070   ierr = PetscMemzero(ca,ci[cm]*sizeof(MatScalar));CHKERRQ(ierr);
1071 
1072   ierr = PetscObjectQuery((PetscObject)C,"Mat_MatMatTransMult",(PetscObject*)&container);CHKERRQ(ierr);
1073   if (!container) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Container does not exit");
1074   ierr = PetscContainerGetPointer(container,(void**)&multtrans);CHKERRQ(ierr);
1075   if (multtrans->usecoloring) {
1076     MatTransposeColoring matcoloring = multtrans->matcoloring;
1077     Mat                  Bt_dense;
1078     PetscInt             m,n;
1079     Mat                  C_dense = multtrans->ABt_den;
1080 
1081     Bt_dense = multtrans->Bt_den;
1082     ierr     = MatGetLocalSize(Bt_dense,&m,&n);CHKERRQ(ierr);
1083 
1084     /* Get Bt_dense by Apply MatTransposeColoring to B */
1085     ierr = MatTransColoringApplySpToDen(matcoloring,B,Bt_dense);CHKERRQ(ierr);
1086 
1087     /* C_dense = A*Bt_dense */
1088     ierr = MatMatMultNumeric_SeqAIJ_SeqDense(A,Bt_dense,C_dense);CHKERRQ(ierr);
1089 
1090     /* Recover C from C_dense */
1091     ierr = MatTransColoringApplyDenToSp(matcoloring,C_dense,C);CHKERRQ(ierr);
1092     PetscFunctionReturn(0);
1093   }
1094 
1095 #if defined(USE_ARRAY)
1096   /* allocate an array for implementing sparse inner-product */
1097   ierr = PetscMalloc((A->cmap->n+1)*sizeof(MatScalar),&spdot);CHKERRQ(ierr);
1098   ierr = PetscMemzero(spdot,(A->cmap->n+1)*sizeof(MatScalar));CHKERRQ(ierr);
1099 #endif
1100 
1101   for (i=0; i<cm; i++) {
1102     anzi = ai[i+1] - ai[i];
1103     acol = aj + ai[i];
1104     aval = aa + ai[i];
1105     cnzi = ci[i+1] - ci[i];
1106     ccol = cj + ci[i];
1107     cval = ca + ci[i];
1108     for (j=0; j<cnzi; j++) {
1109       brow = ccol[j];
1110       bnzj = bi[brow+1] - bi[brow];
1111       bcol = bj + bi[brow];
1112       bval = ba + bi[brow];
1113 
1114       /* perform sparse inner-product c(i,j)=A[i,:]*B[j,:]^T */
1115 #if defined(USE_ARRAY)
1116       /* put ba to spdot array */
1117       for (nextb=0; nextb<bnzj; nextb++) spdot[bcol[nextb]] = bval[nextb];
1118       /* c(i,j)=A[i,:]*B[j,:]^T */
1119       for (nexta=0; nexta<anzi; nexta++) {
1120         cval[j] += spdot[acol[nexta]]*aval[nexta];
1121       }
1122       /* zero spdot array */
1123       for (nextb=0; nextb<bnzj; nextb++) spdot[bcol[nextb]] = 0.0;
1124 #else
1125       nexta = 0; nextb = 0;
1126       while (nexta<anzi && nextb<bnzj) {
1127         while (acol[nexta] < bcol[nextb] && nexta < anzi) nexta++;
1128         if (nexta == anzi) break;
1129         while (acol[nexta] > bcol[nextb] && nextb < bnzj) nextb++;
1130         if (nextb == bnzj) break;
1131         if (acol[nexta] == bcol[nextb]) {
1132           cval[j] += aval[nexta]*bval[nextb];
1133           nexta++; nextb++;
1134           flops += 2;
1135         }
1136       }
1137 #endif
1138     }
1139   }
1140   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1141   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1142   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
1143 #if defined(USE_ARRAY)
1144   ierr = PetscFree(spdot);
1145 #endif
1146   PetscFunctionReturn(0);
1147 }
1148 
1149 #undef __FUNCT__
1150 #define __FUNCT__ "MatTransposeMatMult_SeqAIJ_SeqAIJ"
1151 PetscErrorCode MatTransposeMatMult_SeqAIJ_SeqAIJ(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
1152 {
1153   PetscErrorCode ierr;
1154 
1155   PetscFunctionBegin;
1156   if (scall == MAT_INITIAL_MATRIX) {
1157     ierr = MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ(A,B,fill,C);CHKERRQ(ierr);
1158   }
1159   ierr = MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ(A,B,*C);CHKERRQ(ierr);
1160   PetscFunctionReturn(0);
1161 }
1162 
1163 #undef __FUNCT__
1164 #define __FUNCT__ "MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ"
1165 PetscErrorCode MatTransposeMatMultSymbolic_SeqAIJ_SeqAIJ(Mat A,Mat B,PetscReal fill,Mat *C)
1166 {
1167   PetscErrorCode ierr;
1168   Mat            At;
1169   PetscInt       *ati,*atj;
1170 
1171   PetscFunctionBegin;
1172   /* create symbolic At */
1173   ierr = MatGetSymbolicTranspose_SeqAIJ(A,&ati,&atj);CHKERRQ(ierr);
1174   ierr = MatCreateSeqAIJWithArrays(PETSC_COMM_SELF,A->cmap->n,A->rmap->n,ati,atj,NULL,&At);CHKERRQ(ierr);
1175 
1176   At->rmap->bs = A->cmap->bs;
1177   At->cmap->bs = B->cmap->bs;
1178 
1179   /* get symbolic C=At*B */
1180   ierr = MatMatMultSymbolic_SeqAIJ_SeqAIJ(At,B,fill,C);CHKERRQ(ierr);
1181 
1182   /* clean up */
1183   ierr = MatDestroy(&At);CHKERRQ(ierr);
1184   ierr = MatRestoreSymbolicTranspose_SeqAIJ(A,&ati,&atj);CHKERRQ(ierr);
1185   PetscFunctionReturn(0);
1186 }
1187 
1188 #undef __FUNCT__
1189 #define __FUNCT__ "MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ"
1190 PetscErrorCode MatTransposeMatMultNumeric_SeqAIJ_SeqAIJ(Mat A,Mat B,Mat C)
1191 {
1192   PetscErrorCode ierr;
1193   Mat_SeqAIJ     *a   =(Mat_SeqAIJ*)A->data,*b=(Mat_SeqAIJ*)B->data,*c=(Mat_SeqAIJ*)C->data;
1194   PetscInt       am   =A->rmap->n,anzi,*ai=a->i,*aj=a->j,*bi=b->i,*bj,bnzi,nextb;
1195   PetscInt       cm   =C->rmap->n,*ci=c->i,*cj=c->j,crow,*cjj,i,j,k;
1196   PetscLogDouble flops=0.0;
1197   MatScalar      *aa  =a->a,*ba,*ca,*caj;
1198 
1199   PetscFunctionBegin;
1200   if (!c->a) {
1201     ierr = PetscMalloc((ci[cm]+1)*sizeof(MatScalar),&ca);CHKERRQ(ierr);
1202 
1203     c->a      = ca;
1204     c->free_a = PETSC_TRUE;
1205   } else {
1206     ca = c->a;
1207   }
1208   /* clear old values in C */
1209   ierr = PetscMemzero(ca,ci[cm]*sizeof(MatScalar));CHKERRQ(ierr);
1210 
1211   /* compute A^T*B using outer product (A^T)[:,i]*B[i,:] */
1212   for (i=0; i<am; i++) {
1213     bj   = b->j + bi[i];
1214     ba   = b->a + bi[i];
1215     bnzi = bi[i+1] - bi[i];
1216     anzi = ai[i+1] - ai[i];
1217     for (j=0; j<anzi; j++) {
1218       nextb = 0;
1219       crow  = *aj++;
1220       cjj   = cj + ci[crow];
1221       caj   = ca + ci[crow];
1222       /* perform sparse axpy operation.  Note cjj includes bj. */
1223       for (k=0; nextb<bnzi; k++) {
1224         if (cjj[k] == *(bj+nextb)) { /* ccol == bcol */
1225           caj[k] += (*aa)*(*(ba+nextb));
1226           nextb++;
1227         }
1228       }
1229       flops += 2*bnzi;
1230       aa++;
1231     }
1232   }
1233 
1234   /* Assemble the final matrix and clean up */
1235   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1236   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1237   ierr = PetscLogFlops(flops);CHKERRQ(ierr);
1238   PetscFunctionReturn(0);
1239 }
1240 
1241 #undef __FUNCT__
1242 #define __FUNCT__ "MatMatMult_SeqAIJ_SeqDense"
1243 PetscErrorCode MatMatMult_SeqAIJ_SeqDense(Mat A,Mat B,MatReuse scall,PetscReal fill,Mat *C)
1244 {
1245   PetscErrorCode ierr;
1246 
1247   PetscFunctionBegin;
1248   if (scall == MAT_INITIAL_MATRIX) {
1249     ierr = MatMatMultSymbolic_SeqAIJ_SeqDense(A,B,fill,C);CHKERRQ(ierr);
1250   }
1251   ierr = MatMatMultNumeric_SeqAIJ_SeqDense(A,B,*C);CHKERRQ(ierr);
1252   PetscFunctionReturn(0);
1253 }
1254 
1255 #undef __FUNCT__
1256 #define __FUNCT__ "MatMatMultSymbolic_SeqAIJ_SeqDense"
1257 PetscErrorCode MatMatMultSymbolic_SeqAIJ_SeqDense(Mat A,Mat B,PetscReal fill,Mat *C)
1258 {
1259   PetscErrorCode ierr;
1260 
1261   PetscFunctionBegin;
1262   ierr = MatMatMultSymbolic_SeqDense_SeqDense(A,B,0.0,C);CHKERRQ(ierr);
1263 
1264   (*C)->ops->matmultnumeric = MatMatMultNumeric_SeqAIJ_SeqDense;
1265   PetscFunctionReturn(0);
1266 }
1267 
1268 #undef __FUNCT__
1269 #define __FUNCT__ "MatMatMultNumeric_SeqAIJ_SeqDense"
1270 PetscErrorCode MatMatMultNumeric_SeqAIJ_SeqDense(Mat A,Mat B,Mat C)
1271 {
1272   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
1273   PetscErrorCode ierr;
1274   PetscScalar    *b,*c,r1,r2,r3,r4,*b1,*b2,*b3,*b4;
1275   MatScalar      *aa;
1276   PetscInt       cm  = C->rmap->n, cn=B->cmap->n, bm=B->rmap->n, col, i,j,n,*aj, am = A->rmap->n;
1277   PetscInt       am2 = 2*am, am3 = 3*am,  bm4 = 4*bm,colam;
1278 
1279   PetscFunctionBegin;
1280   if (!cm || !cn) PetscFunctionReturn(0);
1281   if (bm != 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,bm);
1282   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);
1283   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);
1284   ierr = MatDenseGetArray(B,&b);CHKERRQ(ierr);
1285   ierr = MatDenseGetArray(C,&c);CHKERRQ(ierr);
1286   b1   = b; b2 = b1 + bm; b3 = b2 + bm; b4 = b3 + bm;
1287   for (col=0; col<cn-4; col += 4) {  /* over columns of C */
1288     colam = col*am;
1289     for (i=0; i<am; i++) {        /* over rows of C in those columns */
1290       r1 = r2 = r3 = r4 = 0.0;
1291       n  = a->i[i+1] - a->i[i];
1292       aj = a->j + a->i[i];
1293       aa = a->a + a->i[i];
1294       for (j=0; j<n; j++) {
1295         r1 += (*aa)*b1[*aj];
1296         r2 += (*aa)*b2[*aj];
1297         r3 += (*aa)*b3[*aj];
1298         r4 += (*aa++)*b4[*aj++];
1299       }
1300       c[colam + i]       = r1;
1301       c[colam + am + i]  = r2;
1302       c[colam + am2 + i] = r3;
1303       c[colam + am3 + i] = r4;
1304     }
1305     b1 += bm4;
1306     b2 += bm4;
1307     b3 += bm4;
1308     b4 += bm4;
1309   }
1310   for (; col<cn; col++) {     /* over extra columns of C */
1311     for (i=0; i<am; i++) {  /* over rows of C in those columns */
1312       r1 = 0.0;
1313       n  = a->i[i+1] - a->i[i];
1314       aj = a->j + a->i[i];
1315       aa = a->a + a->i[i];
1316 
1317       for (j=0; j<n; j++) {
1318         r1 += (*aa++)*b1[*aj++];
1319       }
1320       c[col*am + i] = r1;
1321     }
1322     b1 += bm;
1323   }
1324   ierr = PetscLogFlops(cn*(2.0*a->nz));CHKERRQ(ierr);
1325   ierr = MatDenseRestoreArray(B,&b);CHKERRQ(ierr);
1326   ierr = MatDenseRestoreArray(C,&c);CHKERRQ(ierr);
1327   ierr = MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1328   ierr = MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
1329   PetscFunctionReturn(0);
1330 }
1331 
1332 /*
1333    Note very similar to MatMult_SeqAIJ(), should generate both codes from same base
1334 */
1335 #undef __FUNCT__
1336 #define __FUNCT__ "MatMatMultNumericAdd_SeqAIJ_SeqDense"
1337 PetscErrorCode MatMatMultNumericAdd_SeqAIJ_SeqDense(Mat A,Mat B,Mat C)
1338 {
1339   Mat_SeqAIJ     *a=(Mat_SeqAIJ*)A->data;
1340   PetscErrorCode ierr;
1341   PetscScalar    *b,*c,r1,r2,r3,r4,*b1,*b2,*b3,*b4;
1342   MatScalar      *aa;
1343   PetscInt       cm  = C->rmap->n, cn=B->cmap->n, bm=B->rmap->n, col, i,j,n,*aj, am = A->rmap->n,*ii,arm;
1344   PetscInt       am2 = 2*am, am3 = 3*am,  bm4 = 4*bm,colam,*ridx;
1345 
1346   PetscFunctionBegin;
1347   if (!cm || !cn) PetscFunctionReturn(0);
1348   ierr = MatDenseGetArray(B,&b);CHKERRQ(ierr);
1349   ierr = MatDenseGetArray(C,&c);CHKERRQ(ierr);
1350   b1   = b; b2 = b1 + bm; b3 = b2 + bm; b4 = b3 + bm;
1351 
1352   if (a->compressedrow.use) { /* use compressed row format */
1353     for (col=0; col<cn-4; col += 4) {  /* over columns of C */
1354       colam = col*am;
1355       arm   = a->compressedrow.nrows;
1356       ii    = a->compressedrow.i;
1357       ridx  = a->compressedrow.rindex;
1358       for (i=0; i<arm; i++) {        /* over rows of C in those columns */
1359         r1 = r2 = r3 = r4 = 0.0;
1360         n  = ii[i+1] - ii[i];
1361         aj = a->j + ii[i];
1362         aa = a->a + ii[i];
1363         for (j=0; j<n; j++) {
1364           r1 += (*aa)*b1[*aj];
1365           r2 += (*aa)*b2[*aj];
1366           r3 += (*aa)*b3[*aj];
1367           r4 += (*aa++)*b4[*aj++];
1368         }
1369         c[colam       + ridx[i]] += r1;
1370         c[colam + am  + ridx[i]] += r2;
1371         c[colam + am2 + ridx[i]] += r3;
1372         c[colam + am3 + ridx[i]] += r4;
1373       }
1374       b1 += bm4;
1375       b2 += bm4;
1376       b3 += bm4;
1377       b4 += bm4;
1378     }
1379     for (; col<cn; col++) {     /* over extra columns of C */
1380       colam = col*am;
1381       arm   = a->compressedrow.nrows;
1382       ii    = a->compressedrow.i;
1383       ridx  = a->compressedrow.rindex;
1384       for (i=0; i<arm; i++) {  /* over rows of C in those columns */
1385         r1 = 0.0;
1386         n  = ii[i+1] - ii[i];
1387         aj = a->j + ii[i];
1388         aa = a->a + ii[i];
1389 
1390         for (j=0; j<n; j++) {
1391           r1 += (*aa++)*b1[*aj++];
1392         }
1393         c[colam + ridx[i]] += r1;
1394       }
1395       b1 += bm;
1396     }
1397   } else {
1398     for (col=0; col<cn-4; col += 4) {  /* over columns of C */
1399       colam = col*am;
1400       for (i=0; i<am; i++) {        /* over rows of C in those columns */
1401         r1 = r2 = r3 = r4 = 0.0;
1402         n  = a->i[i+1] - a->i[i];
1403         aj = a->j + a->i[i];
1404         aa = a->a + a->i[i];
1405         for (j=0; j<n; j++) {
1406           r1 += (*aa)*b1[*aj];
1407           r2 += (*aa)*b2[*aj];
1408           r3 += (*aa)*b3[*aj];
1409           r4 += (*aa++)*b4[*aj++];
1410         }
1411         c[colam + i]       += r1;
1412         c[colam + am + i]  += r2;
1413         c[colam + am2 + i] += r3;
1414         c[colam + am3 + i] += r4;
1415       }
1416       b1 += bm4;
1417       b2 += bm4;
1418       b3 += bm4;
1419       b4 += bm4;
1420     }
1421     for (; col<cn; col++) {     /* over extra columns of C */
1422       colam = col*am;
1423       for (i=0; i<am; i++) {  /* over rows of C in those columns */
1424         r1 = 0.0;
1425         n  = a->i[i+1] - a->i[i];
1426         aj = a->j + a->i[i];
1427         aa = a->a + a->i[i];
1428 
1429         for (j=0; j<n; j++) {
1430           r1 += (*aa++)*b1[*aj++];
1431         }
1432         c[colam + i] += r1;
1433       }
1434       b1 += bm;
1435     }
1436   }
1437   ierr = PetscLogFlops(cn*2.0*a->nz);CHKERRQ(ierr);
1438   ierr = MatDenseRestoreArray(B,&b);CHKERRQ(ierr);
1439   ierr = MatDenseRestoreArray(C,&c);CHKERRQ(ierr);
1440   PetscFunctionReturn(0);
1441 }
1442 
1443 #undef __FUNCT__
1444 #define __FUNCT__ "MatTransColoringApplySpToDen_SeqAIJ"
1445 PetscErrorCode  MatTransColoringApplySpToDen_SeqAIJ(MatTransposeColoring coloring,Mat B,Mat Btdense)
1446 {
1447   PetscErrorCode ierr;
1448   Mat_SeqAIJ     *b       = (Mat_SeqAIJ*)B->data;
1449   Mat_SeqDense   *btdense = (Mat_SeqDense*)Btdense->data;
1450   PetscInt       *bi      = b->i,*bj=b->j;
1451   PetscInt       m        = Btdense->rmap->n,n=Btdense->cmap->n,j,k,l,col,anz,*btcol,brow,ncolumns;
1452   MatScalar      *btval,*btval_den,*ba=b->a;
1453   PetscInt       *columns=coloring->columns,*colorforcol=coloring->colorforcol,ncolors=coloring->ncolors;
1454 
1455   PetscFunctionBegin;
1456   btval_den=btdense->v;
1457   ierr     = PetscMemzero(btval_den,(m*n)*sizeof(MatScalar));CHKERRQ(ierr);
1458   for (k=0; k<ncolors; k++) {
1459     ncolumns = coloring->ncolumns[k];
1460     for (l=0; l<ncolumns; l++) { /* insert a row of B to a column of Btdense */
1461       col   = *(columns + colorforcol[k] + l);
1462       btcol = bj + bi[col];
1463       btval = ba + bi[col];
1464       anz   = bi[col+1] - bi[col];
1465       for (j=0; j<anz; j++) {
1466         brow            = btcol[j];
1467         btval_den[brow] = btval[j];
1468       }
1469     }
1470     btval_den += m;
1471   }
1472   PetscFunctionReturn(0);
1473 }
1474 
1475 #undef __FUNCT__
1476 #define __FUNCT__ "MatTransColoringApplyDenToSp_SeqAIJ"
1477 PetscErrorCode MatTransColoringApplyDenToSp_SeqAIJ(MatTransposeColoring matcoloring,Mat Cden,Mat Csp)
1478 {
1479   PetscErrorCode ierr;
1480   Mat_SeqAIJ     *csp = (Mat_SeqAIJ*)Csp->data;
1481   PetscInt       k,l,*row,*idx,m,ncolors=matcoloring->ncolors,nrows;
1482   PetscScalar    *ca_den,*cp_den,*ca=csp->a;
1483   PetscInt       *rows=matcoloring->rows,*spidx=matcoloring->columnsforspidx,*colorforrow=matcoloring->colorforrow;
1484 
1485   PetscFunctionBegin;
1486   ierr   = MatGetLocalSize(Csp,&m,NULL);CHKERRQ(ierr);
1487   ierr   = MatDenseGetArray(Cden,&ca_den);CHKERRQ(ierr);
1488   cp_den = ca_den;
1489   for (k=0; k<ncolors; k++) {
1490     nrows = matcoloring->nrows[k];
1491     row   = rows  + colorforrow[k];
1492     idx   = spidx + colorforrow[k];
1493     for (l=0; l<nrows; l++) {
1494       ca[idx[l]] = cp_den[row[l]];
1495     }
1496     cp_den += m;
1497   }
1498   ierr = MatDenseRestoreArray(Cden,&ca_den);CHKERRQ(ierr);
1499   PetscFunctionReturn(0);
1500 }
1501 
1502 /*
1503  MatGetColumnIJ_SeqAIJ_Color() and MatRestoreColumnIJ_SeqAIJ_Color() are customized from
1504  MatGetColumnIJ_SeqAIJ() and MatRestoreColumnIJ_SeqAIJ() by adding an output
1505  spidx[], index of a->j, to be used for setting 'columnsforspidx' in MatTransposeColoringCreate_SeqAIJ().
1506  */
1507 #undef __FUNCT__
1508 #define __FUNCT__ "MatGetColumnIJ_SeqAIJ_Color"
1509 PetscErrorCode MatGetColumnIJ_SeqAIJ_Color(Mat A,PetscInt oshift,PetscBool symmetric,PetscBool inodecompressed,PetscInt *nn,const PetscInt *ia[],const PetscInt *ja[],PetscInt *spidx[],PetscBool  *done)
1510 {
1511   Mat_SeqAIJ     *a = (Mat_SeqAIJ*)A->data;
1512   PetscErrorCode ierr;
1513   PetscInt       i,*collengths,*cia,*cja,n = A->cmap->n,m = A->rmap->n;
1514   PetscInt       nz = a->i[m],row,*jj,mr,col;
1515   PetscInt       *cspidx;
1516 
1517   PetscFunctionBegin;
1518   *nn = n;
1519   if (!ia) PetscFunctionReturn(0);
1520   if (symmetric) {
1521     SETERRQ(PetscObjectComm((PetscObject)A),PETSC_ERR_SUP,"MatGetColumnIJ_SeqAIJ_Color() not supported for the case symmetric");
1522     ierr = MatToSymmetricIJ_SeqAIJ(A->rmap->n,a->i,a->j,0,oshift,(PetscInt**)ia,(PetscInt**)ja);CHKERRQ(ierr);
1523   } else {
1524     ierr = PetscMalloc((n+1)*sizeof(PetscInt),&collengths);CHKERRQ(ierr);
1525     ierr = PetscMemzero(collengths,n*sizeof(PetscInt));CHKERRQ(ierr);
1526     ierr = PetscMalloc((n+1)*sizeof(PetscInt),&cia);CHKERRQ(ierr);
1527     ierr = PetscMalloc((nz+1)*sizeof(PetscInt),&cja);CHKERRQ(ierr);
1528     ierr = PetscMalloc((nz+1)*sizeof(PetscInt),&cspidx);CHKERRQ(ierr);
1529     jj   = a->j;
1530     for (i=0; i<nz; i++) {
1531       collengths[jj[i]]++;
1532     }
1533     cia[0] = oshift;
1534     for (i=0; i<n; i++) {
1535       cia[i+1] = cia[i] + collengths[i];
1536     }
1537     ierr = PetscMemzero(collengths,n*sizeof(PetscInt));CHKERRQ(ierr);
1538     jj   = a->j;
1539     for (row=0; row<m; row++) {
1540       mr = a->i[row+1] - a->i[row];
1541       for (i=0; i<mr; i++) {
1542         col = *jj++;
1543 
1544         cspidx[cia[col] + collengths[col] - oshift] = a->i[row] + i; /* index of a->j */
1545         cja[cia[col] + collengths[col]++ - oshift]  = row + oshift;
1546       }
1547     }
1548     ierr   = PetscFree(collengths);CHKERRQ(ierr);
1549     *ia    = cia; *ja = cja;
1550     *spidx = cspidx;
1551   }
1552   PetscFunctionReturn(0);
1553 }
1554 
1555 #undef __FUNCT__
1556 #define __FUNCT__ "MatRestoreColumnIJ_SeqAIJ_Color"
1557 PetscErrorCode MatRestoreColumnIJ_SeqAIJ_Color(Mat A,PetscInt oshift,PetscBool symmetric,PetscBool inodecompressed,PetscInt *n,const PetscInt *ia[],const PetscInt *ja[],PetscInt *spidx[],PetscBool  *done)
1558 {
1559   PetscErrorCode ierr;
1560 
1561   PetscFunctionBegin;
1562   if (!ia) PetscFunctionReturn(0);
1563 
1564   ierr = PetscFree(*ia);CHKERRQ(ierr);
1565   ierr = PetscFree(*ja);CHKERRQ(ierr);
1566   ierr = PetscFree(*spidx);CHKERRQ(ierr);
1567   PetscFunctionReturn(0);
1568 }
1569 
1570 #undef __FUNCT__
1571 #define __FUNCT__ "MatTransposeColoringCreate_SeqAIJ"
1572 PetscErrorCode MatTransposeColoringCreate_SeqAIJ(Mat mat,ISColoring iscoloring,MatTransposeColoring c)
1573 {
1574   PetscErrorCode ierr;
1575   PetscInt       i,n,nrows,N,j,k,m,ncols,col,cm;
1576   const PetscInt *is,*ci,*cj,*row_idx;
1577   PetscInt       nis = iscoloring->n,*rowhit,bs = 1;
1578   IS             *isa;
1579   PetscBool      flg1,flg2;
1580   Mat_SeqAIJ     *csp = (Mat_SeqAIJ*)mat->data;
1581   PetscInt       *colorforrow,*rows,*rows_i,*columnsforspidx,*columnsforspidx_i,*idxhit,*spidx;
1582   PetscInt       *colorforcol,*columns,*columns_i;
1583 
1584   PetscFunctionBegin;
1585   ierr = ISColoringGetIS(iscoloring,PETSC_IGNORE,&isa);CHKERRQ(ierr);
1586 
1587   /* this is ugly way to get blocksize but cannot call MatGetBlockSize() because AIJ can have bs > 1 */
1588   ierr = PetscObjectTypeCompare((PetscObject)mat,MATSEQBAIJ,&flg1);CHKERRQ(ierr);
1589   ierr = PetscObjectTypeCompare((PetscObject)mat,MATMPIBAIJ,&flg2);CHKERRQ(ierr);
1590   if (flg1 || flg2) {
1591     ierr = MatGetBlockSize(mat,&bs);CHKERRQ(ierr);
1592   }
1593 
1594   N         = mat->cmap->N/bs;
1595   c->M      = mat->rmap->N/bs;  /* set total rows, columns and local rows */
1596   c->N      = mat->cmap->N/bs;
1597   c->m      = mat->rmap->N/bs;
1598   c->rstart = 0;
1599 
1600   c->ncolors = nis;
1601   ierr       = PetscMalloc(nis*sizeof(PetscInt),&c->ncolumns);CHKERRQ(ierr);
1602   ierr       = PetscMalloc(nis*sizeof(PetscInt),&c->nrows);CHKERRQ(ierr);
1603   ierr       = PetscMalloc2(csp->nz+1,PetscInt,&rows,csp->nz+1,PetscInt,&columnsforspidx);CHKERRQ(ierr);
1604   ierr       = PetscMalloc((nis+1)*sizeof(PetscInt),&colorforrow);CHKERRQ(ierr);
1605 
1606   colorforrow[0]    = 0;
1607   rows_i            = rows;
1608   columnsforspidx_i = columnsforspidx;
1609 
1610   ierr = PetscMalloc((nis+1)*sizeof(PetscInt),&colorforcol);CHKERRQ(ierr);
1611   ierr = PetscMalloc((N+1)*sizeof(PetscInt),&columns);CHKERRQ(ierr);
1612 
1613   colorforcol[0] = 0;
1614   columns_i      = columns;
1615 
1616   ierr = MatGetColumnIJ_SeqAIJ_Color(mat,0,PETSC_FALSE,PETSC_FALSE,&ncols,&ci,&cj,&spidx,NULL);CHKERRQ(ierr); /* column-wise storage of mat */
1617 
1618   cm   = c->m;
1619   ierr = PetscMalloc((cm+1)*sizeof(PetscInt),&rowhit);CHKERRQ(ierr);
1620   ierr = PetscMalloc((cm+1)*sizeof(PetscInt),&idxhit);CHKERRQ(ierr);
1621   for (i=0; i<nis; i++) {
1622     ierr = ISGetLocalSize(isa[i],&n);CHKERRQ(ierr);
1623     ierr = ISGetIndices(isa[i],&is);CHKERRQ(ierr);
1624 
1625     c->ncolumns[i] = n;
1626     if (n) {
1627       ierr = PetscMemcpy(columns_i,is,n*sizeof(PetscInt));CHKERRQ(ierr);
1628     }
1629     colorforcol[i+1] = colorforcol[i] + n;
1630     columns_i       += n;
1631 
1632     /* fast, crude version requires O(N*N) work */
1633     ierr = PetscMemzero(rowhit,cm*sizeof(PetscInt));CHKERRQ(ierr);
1634 
1635     /* loop over columns*/
1636     for (j=0; j<n; j++) {
1637       col     = is[j];
1638       row_idx = cj + ci[col];
1639       m       = ci[col+1] - ci[col];
1640       /* loop over columns marking them in rowhit */
1641       for (k=0; k<m; k++) {
1642         idxhit[*row_idx]   = spidx[ci[col] + k];
1643         rowhit[*row_idx++] = col + 1;
1644       }
1645     }
1646     /* count the number of hits */
1647     nrows = 0;
1648     for (j=0; j<cm; j++) {
1649       if (rowhit[j]) nrows++;
1650     }
1651     c->nrows[i]      = nrows;
1652     colorforrow[i+1] = colorforrow[i] + nrows;
1653 
1654     nrows = 0;
1655     for (j=0; j<cm; j++) {
1656       if (rowhit[j]) {
1657         rows_i[nrows]            = j;
1658         columnsforspidx_i[nrows] = idxhit[j];
1659         nrows++;
1660       }
1661     }
1662     ierr    = ISRestoreIndices(isa[i],&is);CHKERRQ(ierr);
1663     rows_i += nrows; columnsforspidx_i += nrows;
1664   }
1665   ierr = MatRestoreColumnIJ_SeqAIJ_Color(mat,0,PETSC_FALSE,PETSC_FALSE,&ncols,&ci,&cj,&spidx,NULL);CHKERRQ(ierr);
1666   ierr = PetscFree(rowhit);CHKERRQ(ierr);
1667   ierr = ISColoringRestoreIS(iscoloring,&isa);CHKERRQ(ierr);
1668 #if defined(PETSC_USE_DEBUG)
1669   if (csp->nz != colorforrow[nis]) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_PLIB,"csp->nz %d != colorforrow[nis] %d",csp->nz,colorforrow[nis]);
1670 #endif
1671 
1672   c->colorforrow     = colorforrow;
1673   c->rows            = rows;
1674   c->columnsforspidx = columnsforspidx;
1675   c->colorforcol     = colorforcol;
1676   c->columns         = columns;
1677 
1678   ierr = PetscFree(idxhit);CHKERRQ(ierr);
1679   PetscFunctionReturn(0);
1680 }
1681