xref: /petsc/src/mat/impls/dense/mpi/mmdense.c (revision d736bfeb4d37a01fcbdf00fe73fb60d6f0ba2142)
1 #define PETSCMAT_DLL
2 
3 /*
4    Support for the parallel dense matrix vector multiply
5 */
6 #include "../src/mat/impls/dense/mpi/mpidense.h"
7 #include "petscblaslapack.h"
8 
9 #undef __FUNCT__
10 #define __FUNCT__ "MatSetUpMultiply_MPIDense"
11 PetscErrorCode MatSetUpMultiply_MPIDense(Mat mat)
12 {
13   Mat_MPIDense *mdn = (Mat_MPIDense*)mat->data;
14   PetscErrorCode ierr;
15   IS           from,to;
16   Vec          gvec;
17 
18   PetscFunctionBegin;
19   /* Create local vector that is used to scatter into */
20   ierr = VecCreateSeq(PETSC_COMM_SELF,mat->cmap->N,&mdn->lvec);CHKERRQ(ierr);
21 
22   /* Create temporary index set for building scatter gather */
23   ierr = ISCreateStride(((PetscObject)mat)->comm,mat->cmap->N,0,1,&from);CHKERRQ(ierr);
24   ierr = ISCreateStride(PETSC_COMM_SELF,mat->cmap->N,0,1,&to);CHKERRQ(ierr);
25 
26   /* Create temporary global vector to generate scatter context */
27   /* n    = mdn->cowners[mdn->rank+1] - mdn->cowners[mdn->rank]; */
28 
29   ierr = VecCreateMPIWithArray(((PetscObject)mat)->comm,mdn->nvec,mat->cmap->N,PETSC_NULL,&gvec);CHKERRQ(ierr);
30 
31   /* Generate the scatter context */
32   ierr = VecScatterCreate(gvec,from,mdn->lvec,to,&mdn->Mvctx);CHKERRQ(ierr);
33   ierr = PetscLogObjectParent(mat,mdn->Mvctx);CHKERRQ(ierr);
34   ierr = PetscLogObjectParent(mat,mdn->lvec);CHKERRQ(ierr);
35   ierr = PetscLogObjectParent(mat,from);CHKERRQ(ierr);
36   ierr = PetscLogObjectParent(mat,to);CHKERRQ(ierr);
37   ierr = PetscLogObjectParent(mat,gvec);CHKERRQ(ierr);
38 
39   ierr = ISDestroy(to);CHKERRQ(ierr);
40   ierr = ISDestroy(from);CHKERRQ(ierr);
41   ierr = VecDestroy(gvec);CHKERRQ(ierr);
42   PetscFunctionReturn(0);
43 }
44 
45 EXTERN PetscErrorCode MatGetSubMatrices_MPIDense_Local(Mat,PetscInt,const IS[],const IS[],MatReuse,Mat*);
46 #undef __FUNCT__
47 #define __FUNCT__ "MatGetSubMatrices_MPIDense"
48 PetscErrorCode MatGetSubMatrices_MPIDense(Mat C,PetscInt ismax,const IS isrow[],const IS iscol[],MatReuse scall,Mat *submat[])
49 {
50   PetscErrorCode ierr;
51   PetscInt           nmax,nstages_local,nstages,i,pos,max_no;
52 
53   PetscFunctionBegin;
54   /* Allocate memory to hold all the submatrices */
55   if (scall != MAT_REUSE_MATRIX) {
56     ierr = PetscMalloc((ismax+1)*sizeof(Mat),submat);CHKERRQ(ierr);
57   }
58   /* Determine the number of stages through which submatrices are done */
59   nmax          = 20*1000000 / (C->cmap->N * sizeof(PetscInt));
60   if (!nmax) nmax = 1;
61   nstages_local = ismax/nmax + ((ismax % nmax)?1:0);
62 
63   /* Make sure every processor loops through the nstages */
64   ierr = MPI_Allreduce(&nstages_local,&nstages,1,MPIU_INT,MPI_MAX,((PetscObject)C)->comm);CHKERRQ(ierr);
65 
66 
67   for (i=0,pos=0; i<nstages; i++) {
68     if (pos+nmax <= ismax) max_no = nmax;
69     else if (pos == ismax) max_no = 0;
70     else                   max_no = ismax-pos;
71     ierr = MatGetSubMatrices_MPIDense_Local(C,max_no,isrow+pos,iscol+pos,scall,*submat+pos);CHKERRQ(ierr);
72     pos += max_no;
73   }
74   PetscFunctionReturn(0);
75 }
76 /* -------------------------------------------------------------------------*/
77 #undef __FUNCT__
78 #define __FUNCT__ "MatGetSubMatrices_MPIDense_Local"
79 PetscErrorCode MatGetSubMatrices_MPIDense_Local(Mat C,PetscInt ismax,const IS isrow[],const IS iscol[],MatReuse scall,Mat *submats)
80 {
81   Mat_MPIDense   *c = (Mat_MPIDense*)C->data;
82   Mat            A = c->A;
83   Mat_SeqDense   *a = (Mat_SeqDense*)A->data,*mat;
84   PetscErrorCode ierr;
85   PetscMPIInt    rank,size,tag0,tag1,idex,end,i;
86   PetscInt       N = C->cmap->N,rstart = C->rmap->rstart,count;
87   const PetscInt **irow,**icol,*irow_i;
88   PetscInt       *nrow,*ncol,*w1,*w3,*w4,*rtable,start;
89   PetscInt       **sbuf1,m,j,k,l,ct1,**rbuf1,row,proc;
90   PetscInt       nrqs,msz,**ptr,*ctr,*pa,*tmp,bsz,nrqr;
91   PetscInt       is_no,jmax,**rmap,*rmap_i;
92   PetscInt       ctr_j,*sbuf1_j,*rbuf1_i;
93   MPI_Request    *s_waits1,*r_waits1,*s_waits2,*r_waits2;
94   MPI_Status     *r_status1,*r_status2,*s_status1,*s_status2;
95   MPI_Comm       comm;
96   PetscScalar    **rbuf2,**sbuf2;
97   PetscTruth     sorted;
98 
99   PetscFunctionBegin;
100   comm   = ((PetscObject)C)->comm;
101   tag0   = ((PetscObject)C)->tag;
102   size   = c->size;
103   rank   = c->rank;
104   m      = C->rmap->N;
105 
106   /* Get some new tags to keep the communication clean */
107   ierr = PetscObjectGetNewTag((PetscObject)C,&tag1);CHKERRQ(ierr);
108 
109     /* Check if the col indices are sorted */
110   for (i=0; i<ismax; i++) {
111     ierr = ISSorted(isrow[i],&sorted);CHKERRQ(ierr);
112     if (!sorted) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"ISrow is not sorted");
113     ierr = ISSorted(iscol[i],&sorted);CHKERRQ(ierr);
114     if (!sorted) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"IScol is not sorted");
115   }
116 
117   ierr = PetscMalloc5(ismax,const PetscInt*,&irow,ismax,const PetscInt*,&icol,ismax,PetscInt,&nrow,ismax,PetscInt,&ncol,m,PetscInt,&rtable);CHKERRQ(ierr);
118   for (i=0; i<ismax; i++) {
119     ierr = ISGetIndices(isrow[i],&irow[i]);CHKERRQ(ierr);
120     ierr = ISGetIndices(iscol[i],&icol[i]);CHKERRQ(ierr);
121     ierr = ISGetLocalSize(isrow[i],&nrow[i]);CHKERRQ(ierr);
122     ierr = ISGetLocalSize(iscol[i],&ncol[i]);CHKERRQ(ierr);
123   }
124 
125   /* Create hash table for the mapping :row -> proc*/
126   for (i=0,j=0; i<size; i++) {
127     jmax = C->rmap->range[i+1];
128     for (; j<jmax; j++) {
129       rtable[j] = i;
130     }
131   }
132 
133   /* evaluate communication - mesg to who,length of mesg, and buffer space
134      required. Based on this, buffers are allocated, and data copied into them*/
135   ierr   = PetscMalloc3(2*size,PetscInt,&w1,size,PetscInt,&w3,size,PetscInt,&w4);CHKERRQ(ierr);
136   ierr = PetscMemzero(w1,size*2*sizeof(PetscInt));CHKERRQ(ierr); /* initialize work vector*/
137   ierr = PetscMemzero(w3,size*sizeof(PetscInt));CHKERRQ(ierr); /* initialize work vector*/
138   for (i=0; i<ismax; i++) {
139     ierr   = PetscMemzero(w4,size*sizeof(PetscInt));CHKERRQ(ierr); /* initialize work vector*/
140     jmax   = nrow[i];
141     irow_i = irow[i];
142     for (j=0; j<jmax; j++) {
143       row  = irow_i[j];
144       proc = rtable[row];
145       w4[proc]++;
146     }
147     for (j=0; j<size; j++) {
148       if (w4[j]) { w1[2*j] += w4[j];  w3[j]++;}
149     }
150   }
151 
152   nrqs       = 0;              /* no of outgoing messages */
153   msz        = 0;              /* total mesg length (for all procs) */
154   w1[2*rank] = 0;              /* no mesg sent to self */
155   w3[rank]   = 0;
156   for (i=0; i<size; i++) {
157     if (w1[2*i])  { w1[2*i+1] = 1; nrqs++;} /* there exists a message to proc i */
158   }
159   ierr = PetscMalloc((nrqs+1)*sizeof(PetscInt),&pa);CHKERRQ(ierr); /*(proc -array)*/
160   for (i=0,j=0; i<size; i++) {
161     if (w1[2*i]) { pa[j] = i; j++; }
162   }
163 
164   /* Each message would have a header = 1 + 2*(no of IS) + data */
165   for (i=0; i<nrqs; i++) {
166     j       = pa[i];
167     w1[2*j] += w1[2*j+1] + 2* w3[j];
168     msz     += w1[2*j];
169   }
170   /* Do a global reduction to determine how many messages to expect*/
171   ierr = PetscMaxSum(comm,w1,&bsz,&nrqr);CHKERRQ(ierr);
172 
173   /* Allocate memory for recv buffers . Make sure rbuf1[0] exists by adding 1 to the buffer length */
174   ierr = PetscMalloc((nrqr+1)*sizeof(PetscInt*),&rbuf1);CHKERRQ(ierr);
175   ierr = PetscMalloc(nrqr*bsz*sizeof(PetscInt),&rbuf1[0]);CHKERRQ(ierr);
176   for (i=1; i<nrqr; ++i) rbuf1[i] = rbuf1[i-1] + bsz;
177 
178   /* Post the receives */
179   ierr = PetscMalloc((nrqr+1)*sizeof(MPI_Request),&r_waits1);CHKERRQ(ierr);
180   for (i=0; i<nrqr; ++i) {
181     ierr = MPI_Irecv(rbuf1[i],bsz,MPIU_INT,MPI_ANY_SOURCE,tag0,comm,r_waits1+i);CHKERRQ(ierr);
182   }
183 
184   /* Allocate Memory for outgoing messages */
185   ierr = PetscMalloc4(size,PetscInt*,&sbuf1,size,PetscInt*,&ptr,2*msz,PetscInt,&tmp,size,PetscInt,&ctr);CHKERRQ(ierr);
186   ierr  = PetscMemzero(sbuf1,size*sizeof(PetscInt*));CHKERRQ(ierr);
187   ierr  = PetscMemzero(ptr,size*sizeof(PetscInt*));CHKERRQ(ierr);
188   {
189     PetscInt *iptr = tmp,ict = 0;
190     for (i=0; i<nrqs; i++) {
191       j         = pa[i];
192       iptr     += ict;
193       sbuf1[j]  = iptr;
194       ict       = w1[2*j];
195     }
196   }
197 
198   /* Form the outgoing messages */
199   /* Initialize the header space */
200   for (i=0; i<nrqs; i++) {
201     j           = pa[i];
202     sbuf1[j][0] = 0;
203     ierr        = PetscMemzero(sbuf1[j]+1,2*w3[j]*sizeof(PetscInt));CHKERRQ(ierr);
204     ptr[j]      = sbuf1[j] + 2*w3[j] + 1;
205   }
206 
207   /* Parse the isrow and copy data into outbuf */
208   for (i=0; i<ismax; i++) {
209     ierr = PetscMemzero(ctr,size*sizeof(PetscInt));CHKERRQ(ierr);
210     irow_i = irow[i];
211     jmax   = nrow[i];
212     for (j=0; j<jmax; j++) {  /* parse the indices of each IS */
213       row  = irow_i[j];
214       proc = rtable[row];
215       if (proc != rank) { /* copy to the outgoing buf*/
216         ctr[proc]++;
217         *ptr[proc] = row;
218         ptr[proc]++;
219       }
220     }
221     /* Update the headers for the current IS */
222     for (j=0; j<size; j++) { /* Can Optimise this loop too */
223       if ((ctr_j = ctr[j])) {
224         sbuf1_j        = sbuf1[j];
225         k              = ++sbuf1_j[0];
226         sbuf1_j[2*k]   = ctr_j;
227         sbuf1_j[2*k-1] = i;
228       }
229     }
230   }
231 
232   /*  Now  post the sends */
233   ierr = PetscMalloc((nrqs+1)*sizeof(MPI_Request),&s_waits1);CHKERRQ(ierr);
234   for (i=0; i<nrqs; ++i) {
235     j = pa[i];
236     ierr = MPI_Isend(sbuf1[j],w1[2*j],MPIU_INT,j,tag0,comm,s_waits1+i);CHKERRQ(ierr);
237   }
238 
239   /* Post recieves to capture the row_data from other procs */
240   ierr  = PetscMalloc((nrqs+1)*sizeof(MPI_Request),&r_waits2);CHKERRQ(ierr);
241   ierr  = PetscMalloc((nrqs+1)*sizeof(PetscScalar*),&rbuf2);CHKERRQ(ierr);
242   for (i=0; i<nrqs; i++) {
243     j        = pa[i];
244     count    = (w1[2*j] - (2*sbuf1[j][0] + 1))*N;
245     ierr     = PetscMalloc((count+1)*sizeof(PetscScalar),&rbuf2[i]);CHKERRQ(ierr);
246     ierr     = MPI_Irecv(rbuf2[i],count,MPIU_SCALAR,j,tag1,comm,r_waits2+i);CHKERRQ(ierr);
247   }
248 
249   /* Receive messages(row_nos) and then, pack and send off the rowvalues
250      to the correct processors */
251 
252   ierr = PetscMalloc((nrqr+1)*sizeof(MPI_Request),&s_waits2);CHKERRQ(ierr);
253   ierr = PetscMalloc((nrqr+1)*sizeof(MPI_Status),&r_status1);CHKERRQ(ierr);
254   ierr = PetscMalloc((nrqr+1)*sizeof(PetscScalar*),&sbuf2);CHKERRQ(ierr);
255 
256   {
257     PetscScalar *sbuf2_i,*v_start;
258     PetscInt         s_proc;
259     for (i=0; i<nrqr; ++i) {
260       ierr = MPI_Waitany(nrqr,r_waits1,&idex,r_status1+i);CHKERRQ(ierr);
261       s_proc          = r_status1[i].MPI_SOURCE; /* send processor */
262       rbuf1_i         = rbuf1[idex]; /* Actual message from s_proc */
263       /* no of rows = end - start; since start is array idex[], 0idex, whel end
264          is length of the buffer - which is 1idex */
265       start           = 2*rbuf1_i[0] + 1;
266       ierr            = MPI_Get_count(r_status1+i,MPIU_INT,&end);CHKERRQ(ierr);
267       /* allocate memory sufficinet to hold all the row values */
268       ierr = PetscMalloc((end-start)*N*sizeof(PetscScalar),&sbuf2[idex]);CHKERRQ(ierr);
269       sbuf2_i      = sbuf2[idex];
270       /* Now pack the data */
271       for (j=start; j<end; j++) {
272         row = rbuf1_i[j] - rstart;
273         v_start = a->v + row;
274         for (k=0; k<N; k++) {
275           sbuf2_i[0] = v_start[0];
276           sbuf2_i++; v_start += C->rmap->n;
277         }
278       }
279       /* Now send off the data */
280       ierr = MPI_Isend(sbuf2[idex],(end-start)*N,MPIU_SCALAR,s_proc,tag1,comm,s_waits2+i);CHKERRQ(ierr);
281     }
282   }
283   /* End Send-Recv of IS + row_numbers */
284   ierr = PetscFree(r_status1);CHKERRQ(ierr);
285   ierr = PetscFree(r_waits1);CHKERRQ(ierr);
286   ierr = PetscMalloc((nrqs+1)*sizeof(MPI_Status),&s_status1);CHKERRQ(ierr);
287   if (nrqs) {ierr = MPI_Waitall(nrqs,s_waits1,s_status1);CHKERRQ(ierr);}
288   ierr = PetscFree(s_status1);CHKERRQ(ierr);
289   ierr = PetscFree(s_waits1);CHKERRQ(ierr);
290 
291   /* Create the submatrices */
292   if (scall == MAT_REUSE_MATRIX) {
293     for (i=0; i<ismax; i++) {
294       mat = (Mat_SeqDense *)(submats[i]->data);
295       if ((submats[i]->rmap->n != nrow[i]) || (submats[i]->cmap->n != ncol[i])) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Cannot reuse matrix. wrong size");
296       ierr = PetscMemzero(mat->v,submats[i]->rmap->n*submats[i]->cmap->n*sizeof(PetscScalar));CHKERRQ(ierr);
297       submats[i]->factortype = C->factortype;
298     }
299   } else {
300     for (i=0; i<ismax; i++) {
301       ierr = MatCreate(PETSC_COMM_SELF,submats+i);CHKERRQ(ierr);
302       ierr = MatSetSizes(submats[i],nrow[i],ncol[i],nrow[i],ncol[i]);CHKERRQ(ierr);
303       ierr = MatSetType(submats[i],((PetscObject)A)->type_name);CHKERRQ(ierr);
304       ierr = MatSeqDenseSetPreallocation(submats[i],PETSC_NULL);CHKERRQ(ierr);
305     }
306   }
307 
308   /* Assemble the matrices */
309   {
310     PetscInt         col;
311     PetscScalar *imat_v,*mat_v,*imat_vi,*mat_vi;
312 
313     for (i=0; i<ismax; i++) {
314       mat       = (Mat_SeqDense*)submats[i]->data;
315       mat_v     = a->v;
316       imat_v    = mat->v;
317       irow_i    = irow[i];
318       m         = nrow[i];
319       for (j=0; j<m; j++) {
320         row      = irow_i[j] ;
321         proc     = rtable[row];
322         if (proc == rank) {
323           row      = row - rstart;
324           mat_vi   = mat_v + row;
325           imat_vi  = imat_v + j;
326           for (k=0; k<ncol[i]; k++) {
327             col = icol[i][k];
328             imat_vi[k*m] = mat_vi[col*C->rmap->n];
329           }
330         }
331       }
332     }
333   }
334 
335   /* Create row map-> This maps c->row to submat->row for each submat*/
336   /* this is a very expensive operation wrt memory usage */
337   ierr    = PetscMalloc(ismax*sizeof(PetscInt*),&rmap);CHKERRQ(ierr);
338   ierr    = PetscMalloc(ismax*C->rmap->N*sizeof(PetscInt),&rmap[0]);CHKERRQ(ierr);
339   ierr    = PetscMemzero(rmap[0],ismax*C->rmap->N*sizeof(PetscInt));CHKERRQ(ierr);
340   for (i=1; i<ismax; i++) { rmap[i] = rmap[i-1] + C->rmap->N;}
341   for (i=0; i<ismax; i++) {
342     rmap_i = rmap[i];
343     irow_i = irow[i];
344     jmax   = nrow[i];
345     for (j=0; j<jmax; j++) {
346       rmap_i[irow_i[j]] = j;
347     }
348   }
349 
350   /* Now Receive the row_values and assemble the rest of the matrix */
351   ierr = PetscMalloc((nrqs+1)*sizeof(MPI_Status),&r_status2);CHKERRQ(ierr);
352   {
353     PetscInt    is_max,tmp1,col,*sbuf1_i,is_sz;
354     PetscScalar *rbuf2_i,*imat_v,*imat_vi;
355 
356     for (tmp1=0; tmp1<nrqs; tmp1++) { /* For each message */
357       ierr = MPI_Waitany(nrqs,r_waits2,&i,r_status2+tmp1);CHKERRQ(ierr);
358       /* Now dig out the corresponding sbuf1, which contains the IS data_structure */
359       sbuf1_i = sbuf1[pa[i]];
360       is_max  = sbuf1_i[0];
361       ct1     = 2*is_max+1;
362       rbuf2_i = rbuf2[i];
363       for (j=1; j<=is_max; j++) { /* For each IS belonging to the message */
364         is_no     = sbuf1_i[2*j-1];
365         is_sz     = sbuf1_i[2*j];
366         mat       = (Mat_SeqDense*)submats[is_no]->data;
367         imat_v    = mat->v;
368         rmap_i    = rmap[is_no];
369         m         = nrow[is_no];
370         for (k=0; k<is_sz; k++,rbuf2_i+=N) {  /* For each row */
371           row      = sbuf1_i[ct1]; ct1++;
372           row      = rmap_i[row];
373           imat_vi  = imat_v + row;
374           for (l=0; l<ncol[is_no]; l++) { /* For each col */
375             col = icol[is_no][l];
376             imat_vi[l*m] = rbuf2_i[col];
377           }
378         }
379       }
380     }
381   }
382   /* End Send-Recv of row_values */
383   ierr = PetscFree(r_status2);CHKERRQ(ierr);
384   ierr = PetscFree(r_waits2);CHKERRQ(ierr);
385   ierr = PetscMalloc((nrqr+1)*sizeof(MPI_Status),&s_status2);CHKERRQ(ierr);
386   if (nrqr) {ierr = MPI_Waitall(nrqr,s_waits2,s_status2);CHKERRQ(ierr);}
387   ierr = PetscFree(s_status2);CHKERRQ(ierr);
388   ierr = PetscFree(s_waits2);CHKERRQ(ierr);
389 
390   /* Restore the indices */
391   for (i=0; i<ismax; i++) {
392     ierr = ISRestoreIndices(isrow[i],irow+i);CHKERRQ(ierr);
393     ierr = ISRestoreIndices(iscol[i],icol+i);CHKERRQ(ierr);
394   }
395 
396   /* Destroy allocated memory */
397   ierr = PetscFree5(irow,icol,nrow,ncol,rtable);CHKERRQ(ierr);
398   ierr = PetscFree3(w1,w3,w4);CHKERRQ(ierr);
399   ierr = PetscFree(pa);CHKERRQ(ierr);
400 
401   for (i=0; i<nrqs; ++i) {
402     ierr = PetscFree(rbuf2[i]);CHKERRQ(ierr);
403   }
404   ierr = PetscFree(rbuf2);CHKERRQ(ierr);
405   ierr = PetscFree4(sbuf1,ptr,tmp,ctr);CHKERRQ(ierr);
406   ierr = PetscFree(rbuf1[0]);CHKERRQ(ierr);
407   ierr = PetscFree(rbuf1);CHKERRQ(ierr);
408 
409   for (i=0; i<nrqr; ++i) {
410     ierr = PetscFree(sbuf2[i]);CHKERRQ(ierr);
411   }
412 
413   ierr = PetscFree(sbuf2);CHKERRQ(ierr);
414   ierr = PetscFree(rmap[0]);CHKERRQ(ierr);
415   ierr = PetscFree(rmap);CHKERRQ(ierr);
416 
417   for (i=0; i<ismax; i++) {
418     ierr = MatAssemblyBegin(submats[i],MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
419     ierr = MatAssemblyEnd(submats[i],MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
420   }
421 
422   PetscFunctionReturn(0);
423 }
424 
425 #undef __FUNCT__
426 #define __FUNCT__ "MatScale_MPIDense"
427 PetscErrorCode MatScale_MPIDense(Mat inA,PetscScalar alpha)
428 {
429   Mat_MPIDense   *A = (Mat_MPIDense*)inA->data;
430   Mat_SeqDense   *a = (Mat_SeqDense*)A->A->data;
431   PetscScalar    oalpha = alpha;
432   PetscErrorCode ierr;
433   PetscBLASInt   one = 1,nz = PetscBLASIntCast(inA->rmap->n*inA->cmap->N);
434 
435   PetscFunctionBegin;
436   BLASscal_(&nz,&oalpha,a->v,&one);
437   ierr = PetscLogFlops(nz);CHKERRQ(ierr);
438   PetscFunctionReturn(0);
439 }
440