1 SUBROUTINE DGSVJ1( JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV,
2 $ EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
3 *
4 * -- LAPACK routine (version 3.3.1) --
5 *
6 * -- Contributed by Zlatko Drmac of the University of Zagreb and --
7 * -- Kresimir Veselic of the Fernuniversitaet Hagen --
8 * -- April 2011 --
9 *
10 * -- LAPACK is a software package provided by Univ. of Tennessee, --
11 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
12 *
13 * This routine is also part of SIGMA (version 1.23, October 23. 2008.)
14 * SIGMA is a library of algorithms for highly accurate algorithms for
15 * computation of SVD, PSVD, QSVD, (H,K)-SVD, and for solution of the
16 * eigenvalue problems Hx = lambda M x, H M x = lambda x with H, M > 0.
17 *
18 IMPLICIT NONE
19 * ..
20 * .. Scalar Arguments ..
21 DOUBLE PRECISION EPS, SFMIN, TOL
22 INTEGER INFO, LDA, LDV, LWORK, M, MV, N, N1, NSWEEP
23 CHARACTER*1 JOBV
24 * ..
25 * .. Array Arguments ..
26 DOUBLE PRECISION A( LDA, * ), D( N ), SVA( N ), V( LDV, * ),
27 $ WORK( LWORK )
28 * ..
29 *
30 * Purpose
31 * =======
32 *
33 * DGSVJ1 is called from SGESVJ as a pre-processor and that is its main
34 * purpose. It applies Jacobi rotations in the same way as SGESVJ does, but
35 * it targets only particular pivots and it does not check convergence
36 * (stopping criterion). Few tunning parameters (marked by [TP]) are
37 * available for the implementer.
38 *
39 * Further Details
40 * ~~~~~~~~~~~~~~~
41 * DGSVJ1 applies few sweeps of Jacobi rotations in the column space of
42 * the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
43 * off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
44 * block-entries (tiles) of the (1,2) off-diagonal block are marked by the
45 * [x]'s in the following scheme:
46 *
47 * | * * * [x] [x] [x]|
48 * | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
49 * | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
50 * |[x] [x] [x] * * * |
51 * |[x] [x] [x] * * * |
52 * |[x] [x] [x] * * * |
53 *
54 * In terms of the columns of A, the first N1 columns are rotated 'against'
55 * the remaining N-N1 columns, trying to increase the angle between the
56 * corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
57 * tiled using quadratic tiles of side KBL. Here, KBL is a tunning parmeter.
58 * The number of sweeps is given in NSWEEP and the orthogonality threshold
59 * is given in TOL.
60 *
61 * Contributors
62 * ~~~~~~~~~~~~
63 * Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
64 *
65 * Arguments
66 * =========
67 *
68 * JOBV (input) CHARACTER*1
69 * Specifies whether the output from this procedure is used
70 * to compute the matrix V:
71 * = 'V': the product of the Jacobi rotations is accumulated
72 * by postmulyiplying the N-by-N array V.
73 * (See the description of V.)
74 * = 'A': the product of the Jacobi rotations is accumulated
75 * by postmulyiplying the MV-by-N array V.
76 * (See the descriptions of MV and V.)
77 * = 'N': the Jacobi rotations are not accumulated.
78 *
79 * M (input) INTEGER
80 * The number of rows of the input matrix A. M >= 0.
81 *
82 * N (input) INTEGER
83 * The number of columns of the input matrix A.
84 * M >= N >= 0.
85 *
86 * N1 (input) INTEGER
87 * N1 specifies the 2 x 2 block partition, the first N1 columns are
88 * rotated 'against' the remaining N-N1 columns of A.
89 *
90 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
91 * On entry, M-by-N matrix A, such that A*diag(D) represents
92 * the input matrix.
93 * On exit,
94 * A_onexit * D_onexit represents the input matrix A*diag(D)
95 * post-multiplied by a sequence of Jacobi rotations, where the
96 * rotation threshold and the total number of sweeps are given in
97 * TOL and NSWEEP, respectively.
98 * (See the descriptions of N1, D, TOL and NSWEEP.)
99 *
100 * LDA (input) INTEGER
101 * The leading dimension of the array A. LDA >= max(1,M).
102 *
103 * D (input/workspace/output) DOUBLE PRECISION array, dimension (N)
104 * The array D accumulates the scaling factors from the fast scaled
105 * Jacobi rotations.
106 * On entry, A*diag(D) represents the input matrix.
107 * On exit, A_onexit*diag(D_onexit) represents the input matrix
108 * post-multiplied by a sequence of Jacobi rotations, where the
109 * rotation threshold and the total number of sweeps are given in
110 * TOL and NSWEEP, respectively.
111 * (See the descriptions of N1, A, TOL and NSWEEP.)
112 *
113 * SVA (input/workspace/output) DOUBLE PRECISION array, dimension (N)
114 * On entry, SVA contains the Euclidean norms of the columns of
115 * the matrix A*diag(D).
116 * On exit, SVA contains the Euclidean norms of the columns of
117 * the matrix onexit*diag(D_onexit).
118 *
119 * MV (input) INTEGER
120 * If JOBV .EQ. 'A', then MV rows of V are post-multipled by a
121 * sequence of Jacobi rotations.
122 * If JOBV = 'N', then MV is not referenced.
123 *
124 * V (input/output) DOUBLE PRECISION array, dimension (LDV,N)
125 * If JOBV .EQ. 'V' then N rows of V are post-multipled by a
126 * sequence of Jacobi rotations.
127 * If JOBV .EQ. 'A' then MV rows of V are post-multipled by a
128 * sequence of Jacobi rotations.
129 * If JOBV = 'N', then V is not referenced.
130 *
131 * LDV (input) INTEGER
132 * The leading dimension of the array V, LDV >= 1.
133 * If JOBV = 'V', LDV .GE. N.
134 * If JOBV = 'A', LDV .GE. MV.
135 *
136 * EPS (input) DOUBLE PRECISION
137 * EPS = DLAMCH('Epsilon')
138 *
139 * SFMIN (input) DOUBLE PRECISION
140 * SFMIN = DLAMCH('Safe Minimum')
141 *
142 * TOL (input) DOUBLE PRECISION
143 * TOL is the threshold for Jacobi rotations. For a pair
144 * A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
145 * applied only if DABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.
146 *
147 * NSWEEP (input) INTEGER
148 * NSWEEP is the number of sweeps of Jacobi rotations to be
149 * performed.
150 *
151 * WORK (workspace) DOUBLE PRECISION array, dimension (LWORK)
152 *
153 * LWORK (input) INTEGER
154 * LWORK is the dimension of WORK. LWORK .GE. M.
155 *
156 * INFO (output) INTEGER
157 * = 0 : successful exit.
158 * < 0 : if INFO = -i, then the i-th argument had an illegal value
159 *
160 * =====================================================================
161 *
162 * .. Local Parameters ..
163 DOUBLE PRECISION ZERO, HALF, ONE, TWO
164 PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,
165 $ TWO = 2.0D0 )
166 * ..
167 * .. Local Scalars ..
168 DOUBLE PRECISION AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
169 $ BIGTHETA, CS, LARGE, MXAAPQ, MXSINJ, ROOTBIG,
170 $ ROOTEPS, ROOTSFMIN, ROOTTOL, SMALL, SN, T,
171 $ TEMP1, THETA, THSIGN
172 INTEGER BLSKIP, EMPTSW, i, ibr, igl, IERR, IJBLSK,
173 $ ISWROT, jbc, jgl, KBL, MVL, NOTROT, nblc, nblr,
174 $ p, PSKIPPED, q, ROWSKIP, SWBAND
175 LOGICAL APPLV, ROTOK, RSVEC
176 * ..
177 * .. Local Arrays ..
178 DOUBLE PRECISION FASTR( 5 )
179 * ..
180 * .. Intrinsic Functions ..
181 INTRINSIC DABS, DMAX1, DBLE, MIN0, DSIGN, DSQRT
182 * ..
183 * .. External Functions ..
184 DOUBLE PRECISION DDOT, DNRM2
185 INTEGER IDAMAX
186 LOGICAL LSAME
187 EXTERNAL IDAMAX, LSAME, DDOT, DNRM2
188 * ..
189 * .. External Subroutines ..
190 EXTERNAL DAXPY, DCOPY, DLASCL, DLASSQ, DROTM, DSWAP
191 * ..
192 * .. Executable Statements ..
193 *
194 * Test the input parameters.
195 *
196 APPLV = LSAME( JOBV, 'A' )
197 RSVEC = LSAME( JOBV, 'V' )
198 IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN
199 INFO = -1
200 ELSE IF( M.LT.0 ) THEN
201 INFO = -2
202 ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
203 INFO = -3
204 ELSE IF( N1.LT.0 ) THEN
205 INFO = -4
206 ELSE IF( LDA.LT.M ) THEN
207 INFO = -6
208 ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN
209 INFO = -9
210 ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR.
211 $ ( APPLV.AND.( LDV.LT.MV ) ) ) THEN
212 INFO = -11
213 ELSE IF( TOL.LE.EPS ) THEN
214 INFO = -14
215 ELSE IF( NSWEEP.LT.0 ) THEN
216 INFO = -15
217 ELSE IF( LWORK.LT.M ) THEN
218 INFO = -17
219 ELSE
220 INFO = 0
221 END IF
222 *
223 * #:(
224 IF( INFO.NE.0 ) THEN
225 CALL XERBLA( 'DGSVJ1', -INFO )
226 RETURN
227 END IF
228 *
229 IF( RSVEC ) THEN
230 MVL = N
231 ELSE IF( APPLV ) THEN
232 MVL = MV
233 END IF
234 RSVEC = RSVEC .OR. APPLV
235
236 ROOTEPS = DSQRT( EPS )
237 ROOTSFMIN = DSQRT( SFMIN )
238 SMALL = SFMIN / EPS
239 BIG = ONE / SFMIN
240 ROOTBIG = ONE / ROOTSFMIN
241 LARGE = BIG / DSQRT( DBLE( M*N ) )
242 BIGTHETA = ONE / ROOTEPS
243 ROOTTOL = DSQRT( TOL )
244 *
245 * .. Initialize the right singular vector matrix ..
246 *
247 * RSVEC = LSAME( JOBV, 'Y' )
248 *
249 EMPTSW = N1*( N-N1 )
250 NOTROT = 0
251 FASTR( 1 ) = ZERO
252 *
253 * .. Row-cyclic pivot strategy with de Rijk's pivoting ..
254 *
255 KBL = MIN0( 8, N )
256 NBLR = N1 / KBL
257 IF( ( NBLR*KBL ).NE.N1 )NBLR = NBLR + 1
258
259 * .. the tiling is nblr-by-nblc [tiles]
260
261 NBLC = ( N-N1 ) / KBL
262 IF( ( NBLC*KBL ).NE.( N-N1 ) )NBLC = NBLC + 1
263 BLSKIP = ( KBL**2 ) + 1
264 *[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
265
266 ROWSKIP = MIN0( 5, KBL )
267 *[TP] ROWSKIP is a tuning parameter.
268 SWBAND = 0
269 *[TP] SWBAND is a tuning parameter. It is meaningful and effective
270 * if SGESVJ is used as a computational routine in the preconditioned
271 * Jacobi SVD algorithm SGESVJ.
272 *
273 *
274 * | * * * [x] [x] [x]|
275 * | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
276 * | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
277 * |[x] [x] [x] * * * |
278 * |[x] [x] [x] * * * |
279 * |[x] [x] [x] * * * |
280 *
281 *
282 DO 1993 i = 1, NSWEEP
283 * .. go go go ...
284 *
285 MXAAPQ = ZERO
286 MXSINJ = ZERO
287 ISWROT = 0
288 *
289 NOTROT = 0
290 PSKIPPED = 0
291 *
292 DO 2000 ibr = 1, NBLR
293
294 igl = ( ibr-1 )*KBL + 1
295 *
296 *
297 *........................................................
298 * ... go to the off diagonal blocks
299
300 igl = ( ibr-1 )*KBL + 1
301
302 DO 2010 jbc = 1, NBLC
303
304 jgl = N1 + ( jbc-1 )*KBL + 1
305
306 * doing the block at ( ibr, jbc )
307
308 IJBLSK = 0
309 DO 2100 p = igl, MIN0( igl+KBL-1, N1 )
310
311 AAPP = SVA( p )
312
313 IF( AAPP.GT.ZERO ) THEN
314
315 PSKIPPED = 0
316
317 DO 2200 q = jgl, MIN0( jgl+KBL-1, N )
318 *
319 AAQQ = SVA( q )
320
321 IF( AAQQ.GT.ZERO ) THEN
322 AAPP0 = AAPP
323 *
324 * .. M x 2 Jacobi SVD ..
325 *
326 * .. Safe Gram matrix computation ..
327 *
328 IF( AAQQ.GE.ONE ) THEN
329 IF( AAPP.GE.AAQQ ) THEN
330 ROTOK = ( SMALL*AAPP ).LE.AAQQ
331 ELSE
332 ROTOK = ( SMALL*AAQQ ).LE.AAPP
333 END IF
334 IF( AAPP.LT.( BIG / AAQQ ) ) THEN
335 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
336 $ q ), 1 )*D( p )*D( q ) / AAQQ )
337 $ / AAPP
338 ELSE
339 CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
340 CALL DLASCL( 'G', 0, 0, AAPP, D( p ),
341 $ M, 1, WORK, LDA, IERR )
342 AAPQ = DDOT( M, WORK, 1, A( 1, q ),
343 $ 1 )*D( q ) / AAQQ
344 END IF
345 ELSE
346 IF( AAPP.GE.AAQQ ) THEN
347 ROTOK = AAPP.LE.( AAQQ / SMALL )
348 ELSE
349 ROTOK = AAQQ.LE.( AAPP / SMALL )
350 END IF
351 IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
352 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
353 $ q ), 1 )*D( p )*D( q ) / AAQQ )
354 $ / AAPP
355 ELSE
356 CALL DCOPY( M, A( 1, q ), 1, WORK, 1 )
357 CALL DLASCL( 'G', 0, 0, AAQQ, D( q ),
358 $ M, 1, WORK, LDA, IERR )
359 AAPQ = DDOT( M, WORK, 1, A( 1, p ),
360 $ 1 )*D( p ) / AAPP
361 END IF
362 END IF
363
364 MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) )
365
366 * TO rotate or NOT to rotate, THAT is the question ...
367 *
368 IF( DABS( AAPQ ).GT.TOL ) THEN
369 NOTROT = 0
370 * ROTATED = ROTATED + 1
371 PSKIPPED = 0
372 ISWROT = ISWROT + 1
373 *
374 IF( ROTOK ) THEN
375 *
376 AQOAP = AAQQ / AAPP
377 APOAQ = AAPP / AAQQ
378 THETA = -HALF*DABS(AQOAP-APOAQ) / AAPQ
379 IF( AAQQ.GT.AAPP0 )THETA = -THETA
380
381 IF( DABS( THETA ).GT.BIGTHETA ) THEN
382 T = HALF / THETA
383 FASTR( 3 ) = T*D( p ) / D( q )
384 FASTR( 4 ) = -T*D( q ) / D( p )
385 CALL DROTM( M, A( 1, p ), 1,
386 $ A( 1, q ), 1, FASTR )
387 IF( RSVEC )CALL DROTM( MVL,
388 $ V( 1, p ), 1,
389 $ V( 1, q ), 1,
390 $ FASTR )
391 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
392 $ ONE+T*APOAQ*AAPQ ) )
393 AAPP = AAPP*DSQRT( DMAX1( ZERO,
394 $ ONE-T*AQOAP*AAPQ ) )
395 MXSINJ = DMAX1( MXSINJ, DABS( T ) )
396 ELSE
397 *
398 * .. choose correct signum for THETA and rotate
399 *
400 THSIGN = -DSIGN( ONE, AAPQ )
401 IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
402 T = ONE / ( THETA+THSIGN*
403 $ DSQRT( ONE+THETA*THETA ) )
404 CS = DSQRT( ONE / ( ONE+T*T ) )
405 SN = T*CS
406 MXSINJ = DMAX1( MXSINJ, DABS( SN ) )
407 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
408 $ ONE+T*APOAQ*AAPQ ) )
409 AAPP = AAPP*DSQRT( DMAX1( ZERO,
410 $ ONE-T*AQOAP*AAPQ ) )
411
412 APOAQ = D( p ) / D( q )
413 AQOAP = D( q ) / D( p )
414 IF( D( p ).GE.ONE ) THEN
415 *
416 IF( D( q ).GE.ONE ) THEN
417 FASTR( 3 ) = T*APOAQ
418 FASTR( 4 ) = -T*AQOAP
419 D( p ) = D( p )*CS
420 D( q ) = D( q )*CS
421 CALL DROTM( M, A( 1, p ), 1,
422 $ A( 1, q ), 1,
423 $ FASTR )
424 IF( RSVEC )CALL DROTM( MVL,
425 $ V( 1, p ), 1, V( 1, q ),
426 $ 1, FASTR )
427 ELSE
428 CALL DAXPY( M, -T*AQOAP,
429 $ A( 1, q ), 1,
430 $ A( 1, p ), 1 )
431 CALL DAXPY( M, CS*SN*APOAQ,
432 $ A( 1, p ), 1,
433 $ A( 1, q ), 1 )
434 IF( RSVEC ) THEN
435 CALL DAXPY( MVL, -T*AQOAP,
436 $ V( 1, q ), 1,
437 $ V( 1, p ), 1 )
438 CALL DAXPY( MVL,
439 $ CS*SN*APOAQ,
440 $ V( 1, p ), 1,
441 $ V( 1, q ), 1 )
442 END IF
443 D( p ) = D( p )*CS
444 D( q ) = D( q ) / CS
445 END IF
446 ELSE
447 IF( D( q ).GE.ONE ) THEN
448 CALL DAXPY( M, T*APOAQ,
449 $ A( 1, p ), 1,
450 $ A( 1, q ), 1 )
451 CALL DAXPY( M, -CS*SN*AQOAP,
452 $ A( 1, q ), 1,
453 $ A( 1, p ), 1 )
454 IF( RSVEC ) THEN
455 CALL DAXPY( MVL, T*APOAQ,
456 $ V( 1, p ), 1,
457 $ V( 1, q ), 1 )
458 CALL DAXPY( MVL,
459 $ -CS*SN*AQOAP,
460 $ V( 1, q ), 1,
461 $ V( 1, p ), 1 )
462 END IF
463 D( p ) = D( p ) / CS
464 D( q ) = D( q )*CS
465 ELSE
466 IF( D( p ).GE.D( q ) ) THEN
467 CALL DAXPY( M, -T*AQOAP,
468 $ A( 1, q ), 1,
469 $ A( 1, p ), 1 )
470 CALL DAXPY( M, CS*SN*APOAQ,
471 $ A( 1, p ), 1,
472 $ A( 1, q ), 1 )
473 D( p ) = D( p )*CS
474 D( q ) = D( q ) / CS
475 IF( RSVEC ) THEN
476 CALL DAXPY( MVL,
477 $ -T*AQOAP,
478 $ V( 1, q ), 1,
479 $ V( 1, p ), 1 )
480 CALL DAXPY( MVL,
481 $ CS*SN*APOAQ,
482 $ V( 1, p ), 1,
483 $ V( 1, q ), 1 )
484 END IF
485 ELSE
486 CALL DAXPY( M, T*APOAQ,
487 $ A( 1, p ), 1,
488 $ A( 1, q ), 1 )
489 CALL DAXPY( M,
490 $ -CS*SN*AQOAP,
491 $ A( 1, q ), 1,
492 $ A( 1, p ), 1 )
493 D( p ) = D( p ) / CS
494 D( q ) = D( q )*CS
495 IF( RSVEC ) THEN
496 CALL DAXPY( MVL,
497 $ T*APOAQ, V( 1, p ),
498 $ 1, V( 1, q ), 1 )
499 CALL DAXPY( MVL,
500 $ -CS*SN*AQOAP,
501 $ V( 1, q ), 1,
502 $ V( 1, p ), 1 )
503 END IF
504 END IF
505 END IF
506 END IF
507 END IF
508
509 ELSE
510 IF( AAPP.GT.AAQQ ) THEN
511 CALL DCOPY( M, A( 1, p ), 1, WORK,
512 $ 1 )
513 CALL DLASCL( 'G', 0, 0, AAPP, ONE,
514 $ M, 1, WORK, LDA, IERR )
515 CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
516 $ M, 1, A( 1, q ), LDA,
517 $ IERR )
518 TEMP1 = -AAPQ*D( p ) / D( q )
519 CALL DAXPY( M, TEMP1, WORK, 1,
520 $ A( 1, q ), 1 )
521 CALL DLASCL( 'G', 0, 0, ONE, AAQQ,
522 $ M, 1, A( 1, q ), LDA,
523 $ IERR )
524 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
525 $ ONE-AAPQ*AAPQ ) )
526 MXSINJ = DMAX1( MXSINJ, SFMIN )
527 ELSE
528 CALL DCOPY( M, A( 1, q ), 1, WORK,
529 $ 1 )
530 CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
531 $ M, 1, WORK, LDA, IERR )
532 CALL DLASCL( 'G', 0, 0, AAPP, ONE,
533 $ M, 1, A( 1, p ), LDA,
534 $ IERR )
535 TEMP1 = -AAPQ*D( q ) / D( p )
536 CALL DAXPY( M, TEMP1, WORK, 1,
537 $ A( 1, p ), 1 )
538 CALL DLASCL( 'G', 0, 0, ONE, AAPP,
539 $ M, 1, A( 1, p ), LDA,
540 $ IERR )
541 SVA( p ) = AAPP*DSQRT( DMAX1( ZERO,
542 $ ONE-AAPQ*AAPQ ) )
543 MXSINJ = DMAX1( MXSINJ, SFMIN )
544 END IF
545 END IF
546 * END IF ROTOK THEN ... ELSE
547 *
548 * In the case of cancellation in updating SVA(q)
549 * .. recompute SVA(q)
550 IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
551 $ THEN
552 IF( ( AAQQ.LT.ROOTBIG ) .AND.
553 $ ( AAQQ.GT.ROOTSFMIN ) ) THEN
554 SVA( q ) = DNRM2( M, A( 1, q ), 1 )*
555 $ D( q )
556 ELSE
557 T = ZERO
558 AAQQ = ONE
559 CALL DLASSQ( M, A( 1, q ), 1, T,
560 $ AAQQ )
561 SVA( q ) = T*DSQRT( AAQQ )*D( q )
562 END IF
563 END IF
564 IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
565 IF( ( AAPP.LT.ROOTBIG ) .AND.
566 $ ( AAPP.GT.ROOTSFMIN ) ) THEN
567 AAPP = DNRM2( M, A( 1, p ), 1 )*
568 $ D( p )
569 ELSE
570 T = ZERO
571 AAPP = ONE
572 CALL DLASSQ( M, A( 1, p ), 1, T,
573 $ AAPP )
574 AAPP = T*DSQRT( AAPP )*D( p )
575 END IF
576 SVA( p ) = AAPP
577 END IF
578 * end of OK rotation
579 ELSE
580 NOTROT = NOTROT + 1
581 * SKIPPED = SKIPPED + 1
582 PSKIPPED = PSKIPPED + 1
583 IJBLSK = IJBLSK + 1
584 END IF
585 ELSE
586 NOTROT = NOTROT + 1
587 PSKIPPED = PSKIPPED + 1
588 IJBLSK = IJBLSK + 1
589 END IF
590
591 * IF ( NOTROT .GE. EMPTSW ) GO TO 2011
592 IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) )
593 $ THEN
594 SVA( p ) = AAPP
595 NOTROT = 0
596 GO TO 2011
597 END IF
598 IF( ( i.LE.SWBAND ) .AND.
599 $ ( PSKIPPED.GT.ROWSKIP ) ) THEN
600 AAPP = -AAPP
601 NOTROT = 0
602 GO TO 2203
603 END IF
604
605 *
606 2200 CONTINUE
607 * end of the q-loop
608 2203 CONTINUE
609
610 SVA( p ) = AAPP
611 *
612 ELSE
613 IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
614 $ MIN0( jgl+KBL-1, N ) - jgl + 1
615 IF( AAPP.LT.ZERO )NOTROT = 0
616 *** IF ( NOTROT .GE. EMPTSW ) GO TO 2011
617 END IF
618
619 2100 CONTINUE
620 * end of the p-loop
621 2010 CONTINUE
622 * end of the jbc-loop
623 2011 CONTINUE
624 *2011 bailed out of the jbc-loop
625 DO 2012 p = igl, MIN0( igl+KBL-1, N )
626 SVA( p ) = DABS( SVA( p ) )
627 2012 CONTINUE
628 *** IF ( NOTROT .GE. EMPTSW ) GO TO 1994
629 2000 CONTINUE
630 *2000 :: end of the ibr-loop
631 *
632 * .. update SVA(N)
633 IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) )
634 $ THEN
635 SVA( N ) = DNRM2( M, A( 1, N ), 1 )*D( N )
636 ELSE
637 T = ZERO
638 AAPP = ONE
639 CALL DLASSQ( M, A( 1, N ), 1, T, AAPP )
640 SVA( N ) = T*DSQRT( AAPP )*D( N )
641 END IF
642 *
643 * Additional steering devices
644 *
645 IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR.
646 $ ( ISWROT.LE.N ) ) )SWBAND = i
647
648 IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.DBLE( N )*TOL ) .AND.
649 $ ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN
650 GO TO 1994
651 END IF
652
653 *
654 IF( NOTROT.GE.EMPTSW )GO TO 1994
655
656 1993 CONTINUE
657 * end i=1:NSWEEP loop
658 * #:) Reaching this point means that the procedure has completed the given
659 * number of sweeps.
660 INFO = NSWEEP - 1
661 GO TO 1995
662 1994 CONTINUE
663 * #:) Reaching this point means that during the i-th sweep all pivots were
664 * below the given threshold, causing early exit.
665
666 INFO = 0
667 * #:) INFO = 0 confirms successful iterations.
668 1995 CONTINUE
669 *
670 * Sort the vector D
671 *
672 DO 5991 p = 1, N - 1
673 q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
674 IF( p.NE.q ) THEN
675 TEMP1 = SVA( p )
676 SVA( p ) = SVA( q )
677 SVA( q ) = TEMP1
678 TEMP1 = D( p )
679 D( p ) = D( q )
680 D( q ) = TEMP1
681 CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
682 IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
683 END IF
684 5991 CONTINUE
685 *
686 RETURN
687 * ..
688 * .. END OF DGSVJ1
689 * ..
690 END
2 $ EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
3 *
4 * -- LAPACK routine (version 3.3.1) --
5 *
6 * -- Contributed by Zlatko Drmac of the University of Zagreb and --
7 * -- Kresimir Veselic of the Fernuniversitaet Hagen --
8 * -- April 2011 --
9 *
10 * -- LAPACK is a software package provided by Univ. of Tennessee, --
11 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
12 *
13 * This routine is also part of SIGMA (version 1.23, October 23. 2008.)
14 * SIGMA is a library of algorithms for highly accurate algorithms for
15 * computation of SVD, PSVD, QSVD, (H,K)-SVD, and for solution of the
16 * eigenvalue problems Hx = lambda M x, H M x = lambda x with H, M > 0.
17 *
18 IMPLICIT NONE
19 * ..
20 * .. Scalar Arguments ..
21 DOUBLE PRECISION EPS, SFMIN, TOL
22 INTEGER INFO, LDA, LDV, LWORK, M, MV, N, N1, NSWEEP
23 CHARACTER*1 JOBV
24 * ..
25 * .. Array Arguments ..
26 DOUBLE PRECISION A( LDA, * ), D( N ), SVA( N ), V( LDV, * ),
27 $ WORK( LWORK )
28 * ..
29 *
30 * Purpose
31 * =======
32 *
33 * DGSVJ1 is called from SGESVJ as a pre-processor and that is its main
34 * purpose. It applies Jacobi rotations in the same way as SGESVJ does, but
35 * it targets only particular pivots and it does not check convergence
36 * (stopping criterion). Few tunning parameters (marked by [TP]) are
37 * available for the implementer.
38 *
39 * Further Details
40 * ~~~~~~~~~~~~~~~
41 * DGSVJ1 applies few sweeps of Jacobi rotations in the column space of
42 * the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
43 * off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
44 * block-entries (tiles) of the (1,2) off-diagonal block are marked by the
45 * [x]'s in the following scheme:
46 *
47 * | * * * [x] [x] [x]|
48 * | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
49 * | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
50 * |[x] [x] [x] * * * |
51 * |[x] [x] [x] * * * |
52 * |[x] [x] [x] * * * |
53 *
54 * In terms of the columns of A, the first N1 columns are rotated 'against'
55 * the remaining N-N1 columns, trying to increase the angle between the
56 * corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
57 * tiled using quadratic tiles of side KBL. Here, KBL is a tunning parmeter.
58 * The number of sweeps is given in NSWEEP and the orthogonality threshold
59 * is given in TOL.
60 *
61 * Contributors
62 * ~~~~~~~~~~~~
63 * Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
64 *
65 * Arguments
66 * =========
67 *
68 * JOBV (input) CHARACTER*1
69 * Specifies whether the output from this procedure is used
70 * to compute the matrix V:
71 * = 'V': the product of the Jacobi rotations is accumulated
72 * by postmulyiplying the N-by-N array V.
73 * (See the description of V.)
74 * = 'A': the product of the Jacobi rotations is accumulated
75 * by postmulyiplying the MV-by-N array V.
76 * (See the descriptions of MV and V.)
77 * = 'N': the Jacobi rotations are not accumulated.
78 *
79 * M (input) INTEGER
80 * The number of rows of the input matrix A. M >= 0.
81 *
82 * N (input) INTEGER
83 * The number of columns of the input matrix A.
84 * M >= N >= 0.
85 *
86 * N1 (input) INTEGER
87 * N1 specifies the 2 x 2 block partition, the first N1 columns are
88 * rotated 'against' the remaining N-N1 columns of A.
89 *
90 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
91 * On entry, M-by-N matrix A, such that A*diag(D) represents
92 * the input matrix.
93 * On exit,
94 * A_onexit * D_onexit represents the input matrix A*diag(D)
95 * post-multiplied by a sequence of Jacobi rotations, where the
96 * rotation threshold and the total number of sweeps are given in
97 * TOL and NSWEEP, respectively.
98 * (See the descriptions of N1, D, TOL and NSWEEP.)
99 *
100 * LDA (input) INTEGER
101 * The leading dimension of the array A. LDA >= max(1,M).
102 *
103 * D (input/workspace/output) DOUBLE PRECISION array, dimension (N)
104 * The array D accumulates the scaling factors from the fast scaled
105 * Jacobi rotations.
106 * On entry, A*diag(D) represents the input matrix.
107 * On exit, A_onexit*diag(D_onexit) represents the input matrix
108 * post-multiplied by a sequence of Jacobi rotations, where the
109 * rotation threshold and the total number of sweeps are given in
110 * TOL and NSWEEP, respectively.
111 * (See the descriptions of N1, A, TOL and NSWEEP.)
112 *
113 * SVA (input/workspace/output) DOUBLE PRECISION array, dimension (N)
114 * On entry, SVA contains the Euclidean norms of the columns of
115 * the matrix A*diag(D).
116 * On exit, SVA contains the Euclidean norms of the columns of
117 * the matrix onexit*diag(D_onexit).
118 *
119 * MV (input) INTEGER
120 * If JOBV .EQ. 'A', then MV rows of V are post-multipled by a
121 * sequence of Jacobi rotations.
122 * If JOBV = 'N', then MV is not referenced.
123 *
124 * V (input/output) DOUBLE PRECISION array, dimension (LDV,N)
125 * If JOBV .EQ. 'V' then N rows of V are post-multipled by a
126 * sequence of Jacobi rotations.
127 * If JOBV .EQ. 'A' then MV rows of V are post-multipled by a
128 * sequence of Jacobi rotations.
129 * If JOBV = 'N', then V is not referenced.
130 *
131 * LDV (input) INTEGER
132 * The leading dimension of the array V, LDV >= 1.
133 * If JOBV = 'V', LDV .GE. N.
134 * If JOBV = 'A', LDV .GE. MV.
135 *
136 * EPS (input) DOUBLE PRECISION
137 * EPS = DLAMCH('Epsilon')
138 *
139 * SFMIN (input) DOUBLE PRECISION
140 * SFMIN = DLAMCH('Safe Minimum')
141 *
142 * TOL (input) DOUBLE PRECISION
143 * TOL is the threshold for Jacobi rotations. For a pair
144 * A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
145 * applied only if DABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.
146 *
147 * NSWEEP (input) INTEGER
148 * NSWEEP is the number of sweeps of Jacobi rotations to be
149 * performed.
150 *
151 * WORK (workspace) DOUBLE PRECISION array, dimension (LWORK)
152 *
153 * LWORK (input) INTEGER
154 * LWORK is the dimension of WORK. LWORK .GE. M.
155 *
156 * INFO (output) INTEGER
157 * = 0 : successful exit.
158 * < 0 : if INFO = -i, then the i-th argument had an illegal value
159 *
160 * =====================================================================
161 *
162 * .. Local Parameters ..
163 DOUBLE PRECISION ZERO, HALF, ONE, TWO
164 PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,
165 $ TWO = 2.0D0 )
166 * ..
167 * .. Local Scalars ..
168 DOUBLE PRECISION AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
169 $ BIGTHETA, CS, LARGE, MXAAPQ, MXSINJ, ROOTBIG,
170 $ ROOTEPS, ROOTSFMIN, ROOTTOL, SMALL, SN, T,
171 $ TEMP1, THETA, THSIGN
172 INTEGER BLSKIP, EMPTSW, i, ibr, igl, IERR, IJBLSK,
173 $ ISWROT, jbc, jgl, KBL, MVL, NOTROT, nblc, nblr,
174 $ p, PSKIPPED, q, ROWSKIP, SWBAND
175 LOGICAL APPLV, ROTOK, RSVEC
176 * ..
177 * .. Local Arrays ..
178 DOUBLE PRECISION FASTR( 5 )
179 * ..
180 * .. Intrinsic Functions ..
181 INTRINSIC DABS, DMAX1, DBLE, MIN0, DSIGN, DSQRT
182 * ..
183 * .. External Functions ..
184 DOUBLE PRECISION DDOT, DNRM2
185 INTEGER IDAMAX
186 LOGICAL LSAME
187 EXTERNAL IDAMAX, LSAME, DDOT, DNRM2
188 * ..
189 * .. External Subroutines ..
190 EXTERNAL DAXPY, DCOPY, DLASCL, DLASSQ, DROTM, DSWAP
191 * ..
192 * .. Executable Statements ..
193 *
194 * Test the input parameters.
195 *
196 APPLV = LSAME( JOBV, 'A' )
197 RSVEC = LSAME( JOBV, 'V' )
198 IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN
199 INFO = -1
200 ELSE IF( M.LT.0 ) THEN
201 INFO = -2
202 ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
203 INFO = -3
204 ELSE IF( N1.LT.0 ) THEN
205 INFO = -4
206 ELSE IF( LDA.LT.M ) THEN
207 INFO = -6
208 ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN
209 INFO = -9
210 ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR.
211 $ ( APPLV.AND.( LDV.LT.MV ) ) ) THEN
212 INFO = -11
213 ELSE IF( TOL.LE.EPS ) THEN
214 INFO = -14
215 ELSE IF( NSWEEP.LT.0 ) THEN
216 INFO = -15
217 ELSE IF( LWORK.LT.M ) THEN
218 INFO = -17
219 ELSE
220 INFO = 0
221 END IF
222 *
223 * #:(
224 IF( INFO.NE.0 ) THEN
225 CALL XERBLA( 'DGSVJ1', -INFO )
226 RETURN
227 END IF
228 *
229 IF( RSVEC ) THEN
230 MVL = N
231 ELSE IF( APPLV ) THEN
232 MVL = MV
233 END IF
234 RSVEC = RSVEC .OR. APPLV
235
236 ROOTEPS = DSQRT( EPS )
237 ROOTSFMIN = DSQRT( SFMIN )
238 SMALL = SFMIN / EPS
239 BIG = ONE / SFMIN
240 ROOTBIG = ONE / ROOTSFMIN
241 LARGE = BIG / DSQRT( DBLE( M*N ) )
242 BIGTHETA = ONE / ROOTEPS
243 ROOTTOL = DSQRT( TOL )
244 *
245 * .. Initialize the right singular vector matrix ..
246 *
247 * RSVEC = LSAME( JOBV, 'Y' )
248 *
249 EMPTSW = N1*( N-N1 )
250 NOTROT = 0
251 FASTR( 1 ) = ZERO
252 *
253 * .. Row-cyclic pivot strategy with de Rijk's pivoting ..
254 *
255 KBL = MIN0( 8, N )
256 NBLR = N1 / KBL
257 IF( ( NBLR*KBL ).NE.N1 )NBLR = NBLR + 1
258
259 * .. the tiling is nblr-by-nblc [tiles]
260
261 NBLC = ( N-N1 ) / KBL
262 IF( ( NBLC*KBL ).NE.( N-N1 ) )NBLC = NBLC + 1
263 BLSKIP = ( KBL**2 ) + 1
264 *[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
265
266 ROWSKIP = MIN0( 5, KBL )
267 *[TP] ROWSKIP is a tuning parameter.
268 SWBAND = 0
269 *[TP] SWBAND is a tuning parameter. It is meaningful and effective
270 * if SGESVJ is used as a computational routine in the preconditioned
271 * Jacobi SVD algorithm SGESVJ.
272 *
273 *
274 * | * * * [x] [x] [x]|
275 * | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
276 * | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
277 * |[x] [x] [x] * * * |
278 * |[x] [x] [x] * * * |
279 * |[x] [x] [x] * * * |
280 *
281 *
282 DO 1993 i = 1, NSWEEP
283 * .. go go go ...
284 *
285 MXAAPQ = ZERO
286 MXSINJ = ZERO
287 ISWROT = 0
288 *
289 NOTROT = 0
290 PSKIPPED = 0
291 *
292 DO 2000 ibr = 1, NBLR
293
294 igl = ( ibr-1 )*KBL + 1
295 *
296 *
297 *........................................................
298 * ... go to the off diagonal blocks
299
300 igl = ( ibr-1 )*KBL + 1
301
302 DO 2010 jbc = 1, NBLC
303
304 jgl = N1 + ( jbc-1 )*KBL + 1
305
306 * doing the block at ( ibr, jbc )
307
308 IJBLSK = 0
309 DO 2100 p = igl, MIN0( igl+KBL-1, N1 )
310
311 AAPP = SVA( p )
312
313 IF( AAPP.GT.ZERO ) THEN
314
315 PSKIPPED = 0
316
317 DO 2200 q = jgl, MIN0( jgl+KBL-1, N )
318 *
319 AAQQ = SVA( q )
320
321 IF( AAQQ.GT.ZERO ) THEN
322 AAPP0 = AAPP
323 *
324 * .. M x 2 Jacobi SVD ..
325 *
326 * .. Safe Gram matrix computation ..
327 *
328 IF( AAQQ.GE.ONE ) THEN
329 IF( AAPP.GE.AAQQ ) THEN
330 ROTOK = ( SMALL*AAPP ).LE.AAQQ
331 ELSE
332 ROTOK = ( SMALL*AAQQ ).LE.AAPP
333 END IF
334 IF( AAPP.LT.( BIG / AAQQ ) ) THEN
335 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
336 $ q ), 1 )*D( p )*D( q ) / AAQQ )
337 $ / AAPP
338 ELSE
339 CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
340 CALL DLASCL( 'G', 0, 0, AAPP, D( p ),
341 $ M, 1, WORK, LDA, IERR )
342 AAPQ = DDOT( M, WORK, 1, A( 1, q ),
343 $ 1 )*D( q ) / AAQQ
344 END IF
345 ELSE
346 IF( AAPP.GE.AAQQ ) THEN
347 ROTOK = AAPP.LE.( AAQQ / SMALL )
348 ELSE
349 ROTOK = AAQQ.LE.( AAPP / SMALL )
350 END IF
351 IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
352 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
353 $ q ), 1 )*D( p )*D( q ) / AAQQ )
354 $ / AAPP
355 ELSE
356 CALL DCOPY( M, A( 1, q ), 1, WORK, 1 )
357 CALL DLASCL( 'G', 0, 0, AAQQ, D( q ),
358 $ M, 1, WORK, LDA, IERR )
359 AAPQ = DDOT( M, WORK, 1, A( 1, p ),
360 $ 1 )*D( p ) / AAPP
361 END IF
362 END IF
363
364 MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) )
365
366 * TO rotate or NOT to rotate, THAT is the question ...
367 *
368 IF( DABS( AAPQ ).GT.TOL ) THEN
369 NOTROT = 0
370 * ROTATED = ROTATED + 1
371 PSKIPPED = 0
372 ISWROT = ISWROT + 1
373 *
374 IF( ROTOK ) THEN
375 *
376 AQOAP = AAQQ / AAPP
377 APOAQ = AAPP / AAQQ
378 THETA = -HALF*DABS(AQOAP-APOAQ) / AAPQ
379 IF( AAQQ.GT.AAPP0 )THETA = -THETA
380
381 IF( DABS( THETA ).GT.BIGTHETA ) THEN
382 T = HALF / THETA
383 FASTR( 3 ) = T*D( p ) / D( q )
384 FASTR( 4 ) = -T*D( q ) / D( p )
385 CALL DROTM( M, A( 1, p ), 1,
386 $ A( 1, q ), 1, FASTR )
387 IF( RSVEC )CALL DROTM( MVL,
388 $ V( 1, p ), 1,
389 $ V( 1, q ), 1,
390 $ FASTR )
391 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
392 $ ONE+T*APOAQ*AAPQ ) )
393 AAPP = AAPP*DSQRT( DMAX1( ZERO,
394 $ ONE-T*AQOAP*AAPQ ) )
395 MXSINJ = DMAX1( MXSINJ, DABS( T ) )
396 ELSE
397 *
398 * .. choose correct signum for THETA and rotate
399 *
400 THSIGN = -DSIGN( ONE, AAPQ )
401 IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
402 T = ONE / ( THETA+THSIGN*
403 $ DSQRT( ONE+THETA*THETA ) )
404 CS = DSQRT( ONE / ( ONE+T*T ) )
405 SN = T*CS
406 MXSINJ = DMAX1( MXSINJ, DABS( SN ) )
407 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
408 $ ONE+T*APOAQ*AAPQ ) )
409 AAPP = AAPP*DSQRT( DMAX1( ZERO,
410 $ ONE-T*AQOAP*AAPQ ) )
411
412 APOAQ = D( p ) / D( q )
413 AQOAP = D( q ) / D( p )
414 IF( D( p ).GE.ONE ) THEN
415 *
416 IF( D( q ).GE.ONE ) THEN
417 FASTR( 3 ) = T*APOAQ
418 FASTR( 4 ) = -T*AQOAP
419 D( p ) = D( p )*CS
420 D( q ) = D( q )*CS
421 CALL DROTM( M, A( 1, p ), 1,
422 $ A( 1, q ), 1,
423 $ FASTR )
424 IF( RSVEC )CALL DROTM( MVL,
425 $ V( 1, p ), 1, V( 1, q ),
426 $ 1, FASTR )
427 ELSE
428 CALL DAXPY( M, -T*AQOAP,
429 $ A( 1, q ), 1,
430 $ A( 1, p ), 1 )
431 CALL DAXPY( M, CS*SN*APOAQ,
432 $ A( 1, p ), 1,
433 $ A( 1, q ), 1 )
434 IF( RSVEC ) THEN
435 CALL DAXPY( MVL, -T*AQOAP,
436 $ V( 1, q ), 1,
437 $ V( 1, p ), 1 )
438 CALL DAXPY( MVL,
439 $ CS*SN*APOAQ,
440 $ V( 1, p ), 1,
441 $ V( 1, q ), 1 )
442 END IF
443 D( p ) = D( p )*CS
444 D( q ) = D( q ) / CS
445 END IF
446 ELSE
447 IF( D( q ).GE.ONE ) THEN
448 CALL DAXPY( M, T*APOAQ,
449 $ A( 1, p ), 1,
450 $ A( 1, q ), 1 )
451 CALL DAXPY( M, -CS*SN*AQOAP,
452 $ A( 1, q ), 1,
453 $ A( 1, p ), 1 )
454 IF( RSVEC ) THEN
455 CALL DAXPY( MVL, T*APOAQ,
456 $ V( 1, p ), 1,
457 $ V( 1, q ), 1 )
458 CALL DAXPY( MVL,
459 $ -CS*SN*AQOAP,
460 $ V( 1, q ), 1,
461 $ V( 1, p ), 1 )
462 END IF
463 D( p ) = D( p ) / CS
464 D( q ) = D( q )*CS
465 ELSE
466 IF( D( p ).GE.D( q ) ) THEN
467 CALL DAXPY( M, -T*AQOAP,
468 $ A( 1, q ), 1,
469 $ A( 1, p ), 1 )
470 CALL DAXPY( M, CS*SN*APOAQ,
471 $ A( 1, p ), 1,
472 $ A( 1, q ), 1 )
473 D( p ) = D( p )*CS
474 D( q ) = D( q ) / CS
475 IF( RSVEC ) THEN
476 CALL DAXPY( MVL,
477 $ -T*AQOAP,
478 $ V( 1, q ), 1,
479 $ V( 1, p ), 1 )
480 CALL DAXPY( MVL,
481 $ CS*SN*APOAQ,
482 $ V( 1, p ), 1,
483 $ V( 1, q ), 1 )
484 END IF
485 ELSE
486 CALL DAXPY( M, T*APOAQ,
487 $ A( 1, p ), 1,
488 $ A( 1, q ), 1 )
489 CALL DAXPY( M,
490 $ -CS*SN*AQOAP,
491 $ A( 1, q ), 1,
492 $ A( 1, p ), 1 )
493 D( p ) = D( p ) / CS
494 D( q ) = D( q )*CS
495 IF( RSVEC ) THEN
496 CALL DAXPY( MVL,
497 $ T*APOAQ, V( 1, p ),
498 $ 1, V( 1, q ), 1 )
499 CALL DAXPY( MVL,
500 $ -CS*SN*AQOAP,
501 $ V( 1, q ), 1,
502 $ V( 1, p ), 1 )
503 END IF
504 END IF
505 END IF
506 END IF
507 END IF
508
509 ELSE
510 IF( AAPP.GT.AAQQ ) THEN
511 CALL DCOPY( M, A( 1, p ), 1, WORK,
512 $ 1 )
513 CALL DLASCL( 'G', 0, 0, AAPP, ONE,
514 $ M, 1, WORK, LDA, IERR )
515 CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
516 $ M, 1, A( 1, q ), LDA,
517 $ IERR )
518 TEMP1 = -AAPQ*D( p ) / D( q )
519 CALL DAXPY( M, TEMP1, WORK, 1,
520 $ A( 1, q ), 1 )
521 CALL DLASCL( 'G', 0, 0, ONE, AAQQ,
522 $ M, 1, A( 1, q ), LDA,
523 $ IERR )
524 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
525 $ ONE-AAPQ*AAPQ ) )
526 MXSINJ = DMAX1( MXSINJ, SFMIN )
527 ELSE
528 CALL DCOPY( M, A( 1, q ), 1, WORK,
529 $ 1 )
530 CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
531 $ M, 1, WORK, LDA, IERR )
532 CALL DLASCL( 'G', 0, 0, AAPP, ONE,
533 $ M, 1, A( 1, p ), LDA,
534 $ IERR )
535 TEMP1 = -AAPQ*D( q ) / D( p )
536 CALL DAXPY( M, TEMP1, WORK, 1,
537 $ A( 1, p ), 1 )
538 CALL DLASCL( 'G', 0, 0, ONE, AAPP,
539 $ M, 1, A( 1, p ), LDA,
540 $ IERR )
541 SVA( p ) = AAPP*DSQRT( DMAX1( ZERO,
542 $ ONE-AAPQ*AAPQ ) )
543 MXSINJ = DMAX1( MXSINJ, SFMIN )
544 END IF
545 END IF
546 * END IF ROTOK THEN ... ELSE
547 *
548 * In the case of cancellation in updating SVA(q)
549 * .. recompute SVA(q)
550 IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
551 $ THEN
552 IF( ( AAQQ.LT.ROOTBIG ) .AND.
553 $ ( AAQQ.GT.ROOTSFMIN ) ) THEN
554 SVA( q ) = DNRM2( M, A( 1, q ), 1 )*
555 $ D( q )
556 ELSE
557 T = ZERO
558 AAQQ = ONE
559 CALL DLASSQ( M, A( 1, q ), 1, T,
560 $ AAQQ )
561 SVA( q ) = T*DSQRT( AAQQ )*D( q )
562 END IF
563 END IF
564 IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
565 IF( ( AAPP.LT.ROOTBIG ) .AND.
566 $ ( AAPP.GT.ROOTSFMIN ) ) THEN
567 AAPP = DNRM2( M, A( 1, p ), 1 )*
568 $ D( p )
569 ELSE
570 T = ZERO
571 AAPP = ONE
572 CALL DLASSQ( M, A( 1, p ), 1, T,
573 $ AAPP )
574 AAPP = T*DSQRT( AAPP )*D( p )
575 END IF
576 SVA( p ) = AAPP
577 END IF
578 * end of OK rotation
579 ELSE
580 NOTROT = NOTROT + 1
581 * SKIPPED = SKIPPED + 1
582 PSKIPPED = PSKIPPED + 1
583 IJBLSK = IJBLSK + 1
584 END IF
585 ELSE
586 NOTROT = NOTROT + 1
587 PSKIPPED = PSKIPPED + 1
588 IJBLSK = IJBLSK + 1
589 END IF
590
591 * IF ( NOTROT .GE. EMPTSW ) GO TO 2011
592 IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) )
593 $ THEN
594 SVA( p ) = AAPP
595 NOTROT = 0
596 GO TO 2011
597 END IF
598 IF( ( i.LE.SWBAND ) .AND.
599 $ ( PSKIPPED.GT.ROWSKIP ) ) THEN
600 AAPP = -AAPP
601 NOTROT = 0
602 GO TO 2203
603 END IF
604
605 *
606 2200 CONTINUE
607 * end of the q-loop
608 2203 CONTINUE
609
610 SVA( p ) = AAPP
611 *
612 ELSE
613 IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
614 $ MIN0( jgl+KBL-1, N ) - jgl + 1
615 IF( AAPP.LT.ZERO )NOTROT = 0
616 *** IF ( NOTROT .GE. EMPTSW ) GO TO 2011
617 END IF
618
619 2100 CONTINUE
620 * end of the p-loop
621 2010 CONTINUE
622 * end of the jbc-loop
623 2011 CONTINUE
624 *2011 bailed out of the jbc-loop
625 DO 2012 p = igl, MIN0( igl+KBL-1, N )
626 SVA( p ) = DABS( SVA( p ) )
627 2012 CONTINUE
628 *** IF ( NOTROT .GE. EMPTSW ) GO TO 1994
629 2000 CONTINUE
630 *2000 :: end of the ibr-loop
631 *
632 * .. update SVA(N)
633 IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) )
634 $ THEN
635 SVA( N ) = DNRM2( M, A( 1, N ), 1 )*D( N )
636 ELSE
637 T = ZERO
638 AAPP = ONE
639 CALL DLASSQ( M, A( 1, N ), 1, T, AAPP )
640 SVA( N ) = T*DSQRT( AAPP )*D( N )
641 END IF
642 *
643 * Additional steering devices
644 *
645 IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR.
646 $ ( ISWROT.LE.N ) ) )SWBAND = i
647
648 IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.DBLE( N )*TOL ) .AND.
649 $ ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN
650 GO TO 1994
651 END IF
652
653 *
654 IF( NOTROT.GE.EMPTSW )GO TO 1994
655
656 1993 CONTINUE
657 * end i=1:NSWEEP loop
658 * #:) Reaching this point means that the procedure has completed the given
659 * number of sweeps.
660 INFO = NSWEEP - 1
661 GO TO 1995
662 1994 CONTINUE
663 * #:) Reaching this point means that during the i-th sweep all pivots were
664 * below the given threshold, causing early exit.
665
666 INFO = 0
667 * #:) INFO = 0 confirms successful iterations.
668 1995 CONTINUE
669 *
670 * Sort the vector D
671 *
672 DO 5991 p = 1, N - 1
673 q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
674 IF( p.NE.q ) THEN
675 TEMP1 = SVA( p )
676 SVA( p ) = SVA( q )
677 SVA( q ) = TEMP1
678 TEMP1 = D( p )
679 D( p ) = D( q )
680 D( q ) = TEMP1
681 CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
682 IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
683 END IF
684 5991 CONTINUE
685 *
686 RETURN
687 * ..
688 * .. END OF DGSVJ1
689 * ..
690 END