1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
|
program pzndrv1
c
c Message Passing Layer: BLACS
c
c Example program to illustrate the idea of reverse communication
c for a standard complex nonsymmetric eigenvalue problem.
c
c We implement example one of ex-complex.doc in DOCUMENTS directory
c
c\Example-1
c ... Suppose we want to solve A*x = lambda*x in regular mode,
c where A is obtained from the standard central difference
c discretization of the convection-diffusion operator
c (Laplacian u) + rho*(du / dx)
c in the domain omega = (0,1)x(0,1), with
c u = 0 on the boundary of omega.
c
c ... OP = A and B = I.
c ... Assume "call av (comm, nloc, nx, mv_buf, x, y)" computes y = A*x
c ... Use mode 1 of PZNAUPD.
c
c\BeginLib
c
c\Routines called
c pznaupd Parallel ARPACK reverse communication interface routine.
c pzneupd Parallel ARPACK routine that returns Ritz values and (optionally)
c Ritz vectors.
c pdznorm2 Parallel version of Level 1 BLAS that computes the norm of a complex vector.
c zaxpy Level 1 BLAS that computes y <- alpha*x+y.
c av Distributed matrix vector multiplication routine that computes A*x.
c tv Matrix vector multiplication routine that computes T*x,
c where T is a tridiagonal matrix. It is used in routine
c av.
c
c\Author
c Richard Lehoucq
c Danny Sorensen
c Chao Yang
c Dept. of Computational &
c Applied Mathematics
c Rice University
c Houston, Texas
c
c\Parallel Modifications
c Kristi Maschhoff
c
c\Revision history:
c Starting Point: Complex Code FILE: ndrv1.F SID: 2.1
c
c\SCCS Information:
c FILE: ndrv1.F SID: 1.1 DATE OF SID: 8/13/96 RELEASE: 1
c
c\Remarks
c 1. None
c
c\EndLib
c---------------------------------------------------------------------------
c
include 'debug.h'
include 'stat.h'
c
c %-----------------%
c | BLACS INTERFACE |
c %-----------------%
c
integer comm, iam, nprocs, nloc,
& nprow, npcol, myprow, mypcol
c
external BLACS_PINFO, BLACS_SETUP, BLACS_GET,
& BLACS_GRIDINIT, BLACS_GRIDINFO
c %-----------------------------%
c | Define maximum dimensions |
c | for all arrays. |
c | MAXN: Maximum dimension |
c | of the A allowed. |
c | MAXNEV: Maximum NEV allowed |
c | MAXNCV: Maximum NCV allowed |
c %-----------------------------%
c
integer maxn, maxnev, maxncv, ldv
parameter (maxn=256, maxnev=12, maxncv=30, ldv=maxn)
c
c %--------------%
c | Local Arrays |
c %--------------%
c
integer iparam(11), ipntr(14)
logical select(maxncv)
Complex*16
& ax(maxn), d(maxncv),
& v(ldv,maxncv), workd(3*maxn),
& workev(3*maxncv), resid(maxn),
& workl(3*maxncv*maxncv+5*maxncv)
Double precision
& rwork(maxncv), rd(maxncv,3)
c
c %---------------%
c | Local Scalars |
c %---------------%
c
character bmat*1, which*2
integer ido, n, nx, nev, ncv, lworkl, info, j,
& ierr, nconv, maxitr, ishfts, mode
Complex*16
& sigma
Double precision
& tol
logical rvec
c
c %----------------------------------------------%
c | Local Buffers needed for BLACS communication |
c %----------------------------------------------%
c
Complex*16
& mv_buf(maxn)
c
c %-----------------------------%
c | BLAS & LAPACK routines used |
c %-----------------------------%
c
Double precision
& pdznorm2
external pdznorm2, zaxpy
c
c %-----------------------%
c | Executable Statements |
c %-----------------------%
c
call BLACS_PINFO( iam, nprocs )
c
c If in PVM, create virtual machine if it doesn't exist
c
if (nprocs .lt. 1) then
if (iam .eq. 0) then
write(*,1000)
read(*, 2000) nprocs
endif
call BLACS_SETUP( iam, nprocs )
endif
c
1000 format('How many processes in machine?')
2000 format(I3)
c
c Set up processors in 1D Grid
c
nprow = nprocs
npcol = 1
c
c Get default system context, and define grid
c
call BLACS_GET( 0, 0, comm )
call BLACS_GRIDINIT( comm, 'Row', nprow, npcol )
call BLACS_GRIDINFO( comm, nprow, npcol, myprow, mypcol )
c
c If I'm not in grid, go to end of program
c
if ( (myprow .ge. nprow) .or. (mypcol .ge. npcol) ) goto 9000
c
ndigit = -3
logfil = 6
mcaupd = 1
c
c %--------------------------------------------------%
c | The number NX is the number of interior points |
c | in the discretization of the 2-dimensional |
c | convection-diffusion operator on the unit |
c | square with zero Dirichlet boundary condition. |
c | The number N(=NX*NX) is the dimension of the |
c | matrix. A standard eigenvalue problem is |
c | solved (BMAT = 'I'). NEV is the number of |
c | eigenvalues to be approximated. The user can |
c | modify NX, NEV, NCV, WHICH to solve problems of |
c | different sizes, and to get different parts of |
c | the spectrum. However, The following |
c | conditions must be satisfied: |
c | N <= MAXN |
c | NEV <= MAXNEV |
c | NEV + 2 <= NCV <= MAXNCV |
c %--------------------------------------------------%
c
nx = 10
n = nx*nx
nev = 4
ncv = 20
c
c %--------------------------------------%
c | Set up distribution of data to nodes |
c %--------------------------------------%
c
nloc = (nx / nprocs)*nx
if ( mod(nx, nprocs) .gt. myprow ) nloc = nloc + nx
c
if ( nloc .gt. maxn ) then
print *, ' ERROR with _NDRV1: NLOC is greater than MAXN '
go to 9000
else if ( nev .gt. maxnev ) then
print *, ' ERROR with _NDRV1: NEV is greater than MAXNEV '
go to 9000
else if ( ncv .gt. maxncv ) then
print *, ' ERROR with _NDRV1: NCV is greater than MAXNCV '
go to 9000
end if
bmat = 'I'
which = 'LM'
c
c %---------------------------------------------------%
c | The work array WORKL is used in ZNAUPD as |
c | workspace. Its dimension LWORKL is set as |
c | illustrated below. The parameter TOL determines |
c | the stopping criterion. If TOL<=0, machine |
c | precision is used. The variable IDO is used for |
c | reverse communication, and is initially set to 0. |
c | Setting INFO=0 indicates that a random vector is |
c | generated to start the ARNOLDI iteration. |
c %---------------------------------------------------%
c
lworkl = 3*ncv**2+5*ncv
tol = 0.0
ido = 0
info = 0
c
c %---------------------------------------------------%
c | This program uses exact shift with respect to |
c | the current Hessenberg matrix (IPARAM(1) = 1). |
c | IPARAM(3) specifies the maximum number of Arnoldi |
c | iterations allowed. Mode 1 of ZNAUPD is used |
c | (IPARAM(7) = 1). All these options can be changed |
c | by the user. For details see the documentation in |
c | ZNAUPD. |
c %---------------------------------------------------%
c
ishfts = 1
maxitr = 300
mode = 1
c
iparam(1) = ishfts
iparam(3) = maxitr
iparam(7) = mode
c
c %-------------------------------------------%
c | M A I N L O O P (Reverse communication) |
c %-------------------------------------------%
c
10 continue
c
c %---------------------------------------------%
c | Repeatedly call the routine ZNAUPD and take |
c | actions indicated by parameter IDO until |
c | either convergence is indicated or maxitr |
c | has been exceeded. |
c %---------------------------------------------%
c
call pznaupd ( comm, ido, bmat, nloc, which,
& nev, tol, resid, ncv, v, ldv, iparam, ipntr,
& workd, workl, lworkl, rwork,info )
c
if (ido .eq. -1 .or. ido .eq. 1) then
c
c %-------------------------------------------%
c | Perform matrix vector multiplication |
c | y <--- OP*x |
c | The user should supply his/her own |
c | matrix vector multiplication routine here |
c | that takes workd(ipntr(1)) as the input |
c | vector, and return the matrix vector |
c | product to workd(ipntr(2)). |
c %-------------------------------------------%
c
call av ( comm, nloc, nx, mv_buf,
& workd(ipntr(1)), workd(ipntr(2)))
c
c %-----------------------------------------%
c | L O O P B A C K to call ZNAUPD again. |
c %-----------------------------------------%
c
go to 10
end if
c
c %----------------------------------------%
c | Either we have convergence or there is |
c | an error. |
c %----------------------------------------%
c
if ( info .lt. 0 ) then
c
c %--------------------------%
c | Error message, check the |
c | documentation in ZNAUPD |
c %--------------------------%
c
if ( myprow .eq. 0 ) then
print *, ' '
print *, ' Error with _naupd, info = ', info
print *, ' Check the documentation of _naupd'
print *, ' '
endif
c
else
c
c %-------------------------------------------%
c | No fatal errors occurred. |
c | Post-Process using ZNEUPD. |
c | |
c | Computed eigenvalues may be extracted. |
c | |
c | Eigenvectors may also be computed now if |
c | desired. (indicated by rvec = .true.) |
c %-------------------------------------------%
c
rvec = .true.
c
call pzneupd (comm, rvec, 'A', select, d, v, ldv, sigma,
& workev, bmat, nloc, which, nev, tol, resid, ncv,
& v, ldv, iparam, ipntr, workd, workl, lworkl,
& rwork, ierr)
c
c %----------------------------------------------%
c | Eigenvalues are returned in the one |
c | dimensional array D. The corresponding |
c | eigenvectors are returned in the first NCONV |
c | (=IPARAM(5)) columns of the two dimensional |
c | array V if requested. Otherwise, an |
c | orthogonal basis for the invariant subspace |
c | corresponding to the eigenvalues in D is |
c | returned in V. |
c %----------------------------------------------%
c
if ( ierr .ne. 0) then
c
c %------------------------------------%
c | Error condition: |
c | Check the documentation of ZNEUPD. |
c %------------------------------------%
c
if ( myprow .eq. 0 ) then
print *, ' '
print *, ' Error with _neupd, info = ', ierr
print *, ' Check the documentation of _neupd. '
print *, ' '
endif
c
else
c
nconv = iparam(5)
do 20 j=1, nconv
c
c %---------------------------%
c | Compute the residual norm |
c | |
c | || A*x - lambda*x || |
c | |
c | for the NCONV accurately |
c | computed eigenvalues and |
c | eigenvectors. (iparam(5) |
c | indicates how many are |
c | accurate to the requested |
c | tolerance) |
c %---------------------------%
c
call av(comm, nloc, nx, mv_buf, v(1,j), ax)
call zaxpy(nloc, -d(j), v(1,j), 1, ax, 1)
rd(j,1) = dble(d(j))
rd(j,2) = dimag(d(j))
rd(j,3) = pdznorm2(comm, nloc, ax, 1)
c
20 continue
c
c %-----------------------------%
c | Display computed residuals. |
c %-----------------------------%
c
call pdmout(comm, 6, nconv, 3, rd, maxncv, -6,
& 'Ritz values (Real, Imag) and direct residuals')
end if
c
c %-------------------------------------------%
c | Print additional convergence information. |
c %-------------------------------------------%
c
if (myprow .eq. 0)then
if ( info .eq. 1) then
print *, ' '
print *, ' Maximum number of iterations reached.'
print *, ' '
else if ( info .eq. 3) then
print *, ' '
print *, ' No shifts could be applied during implicit
& Arnoldi update, try increasing NCV.'
print *, ' '
end if
c
print *, ' '
print *, '_NDRV1'
print *, '====== '
print *, ' '
print *, ' Size of the matrix is ', n
print *, ' The number of processors is ', nprocs
print *, ' The number of Ritz values requested is ', nev
print *, ' The number of Arnoldi vectors generated',
& ' (NCV) is ', ncv
print *, ' What portion of the spectrum: ', which
print *, ' The number of converged Ritz values is ',
& nconv
print *, ' The number of Implicit Arnoldi update',
& ' iterations taken is ', iparam(3)
print *, ' The number of OP*x is ', iparam(9)
print *, ' The convergence criterion is ', tol
print *, ' '
c
endif
end if
c
c %----------------------------%
c | Done with program pzndrv1. |
c %----------------------------%
c
9000 continue
c
c
c %-------------------------%
c | Release resources BLACS |
c %-------------------------%
c
call BLACS_GRIDEXIT ( comm )
call BLACS_EXIT(0)
c
end
c
c==========================================================================
c
c parallel matrix vector subroutine
c
c The matrix used is the convection-diffusion operator
c discretized using centered difference.
c
c Computes w <--- OP*v, where OP is the nx*nx by nx*nx block
c tridiagonal matrix
c
c | T -I |
c |-I T -I |
c OP = | -I T |
c | ... -I|
c | -I T|
c
c derived from the standard central difference discretization
c of the convection-diffusion operator (Laplacian u) + rho*(du/dx)
c with zero boundary condition.
c
c The subroutine TV is called to computed y<---T*x.
c
c----------------------------------------------------------------------------
subroutine av (comm, nloc, nx, mv_buf, v, w )
c
c .. BLACS Declarations ...
integer comm, nprow, npcol, myprow, mypcol
external BLACS_GRIDINFO, zgesd2d, zgerv2d
c
integer nloc, nx, np, j, lo, next, prev
Complex*16
& v(nloc), w(nloc), mv_buf(nx), one
parameter (one = (1.0, 0.0))
external zaxpy, tv
c
call BLACS_GRIDINFO( comm, nprow, npcol, myprow, mypcol )
c
np = nloc/nx
call tv(nx,v(1),w(1))
call zaxpy(nx, -one, v(nx+1), 1, w(1), 1)
c
do 10 j = 2, np-1
lo = (j-1)*nx
call tv(nx, v(lo+1), w(lo+1))
call zaxpy(nx, -one, v(lo-nx+1), 1, w(lo+1), 1)
call zaxpy(nx, -one, v(lo+nx+1), 1, w(lo+1), 1)
10 continue
c
lo = (np-1)*nx
call tv(nx, v(lo+1), w(lo+1))
call zaxpy(nx, -one, v(lo-nx+1), 1, w(lo+1), 1)
c
next = myprow + 1
prev = myprow - 1
if ( myprow .lt. nprow-1 ) then
call zgesd2d( comm, nx, 1, v((np-1)*nx+1), nx, next, mypcol)
endif
if ( myprow .gt. 0 ) then
call zgerv2d( comm, nx, 1, mv_buf, nx, prev, mypcol )
call zaxpy( nx, -one, mv_buf, 1, w(1), 1 )
endif
c
if ( myprow .gt. 0 ) then
call zgesd2d( comm, nx, 1, v(1), nx, prev, mypcol)
endif
if ( myprow .lt. nprow-1 ) then
call zgerv2d( comm, nx, 1, mv_buf, nx, next, mypcol )
call zaxpy( nx, -one, mv_buf, 1, w(lo+1), 1 )
endif
c
return
end
c=========================================================================
subroutine tv (nx, x, y)
c
integer nx, j
Complex*16
& x(nx), y(nx), h, dd, dl, du
c
Complex*16
& one, rho
parameter (one = (1.0, 0.0), rho = (100.0, 0.0))
c
c Compute the matrix vector multiplication y<---T*x
c where T is a nx by nx tridiagonal matrix with DD on the
c diagonal, DL on the subdiagonal, and DU on the superdiagonal
c
h = one / dcmplx(nx+1)
dd = (4.0, 0.0)
dl = -one - (0.5, 0.0)*rho*h
du = -one + (0.5, 0.0)*rho*h
c
y(1) = dd*x(1) + du*x(2)
do 10 j = 2,nx-1
y(j) = dl*x(j-1) + dd*x(j) + du*x(j+1)
10 continue
y(nx) = dl*x(nx-1) + dd*x(nx)
return
end
|
