1 SUBROUTINE DGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS,
2 $ 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 INTEGER INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP
22 DOUBLE PRECISION EPS, SFMIN, TOL
23 CHARACTER*1 JOBV
24 * ..
25 * .. Array Arguments ..
26 DOUBLE PRECISION A( LDA, * ), SVA( N ), D( N ), V( LDV, * ),
27 $ WORK( LWORK )
28 * ..
29 *
30 * Purpose
31 * =======
32 *
33 * DGSVJ0 is called from DGESVJ as a pre-processor and that is its main
34 * purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
35 * it does not check convergence (stopping criterion). Few tuning
36 * parameters (marked by [TP]) are available for the implementer.
37 *
38 * Further Details
39 * ~~~~~~~~~~~~~~~
40 * DGSVJ0 is used just to enable SGESVJ to call a simplified version of
41 * itself to work on a submatrix of the original matrix.
42 *
43 * Contributors
44 * ~~~~~~~~~~~~
45 * Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
46 *
47 * Bugs, Examples and Comments
48 * ~~~~~~~~~~~~~~~~~~~~~~~~~~~
49 * Please report all bugs and send interesting test examples and comments to
50 * drmac@math.hr. Thank you.
51 *
52 * Arguments
53 * =========
54 *
55 * JOBV (input) CHARACTER*1
56 * Specifies whether the output from this procedure is used
57 * to compute the matrix V:
58 * = 'V': the product of the Jacobi rotations is accumulated
59 * by postmulyiplying the N-by-N array V.
60 * (See the description of V.)
61 * = 'A': the product of the Jacobi rotations is accumulated
62 * by postmulyiplying the MV-by-N array V.
63 * (See the descriptions of MV and V.)
64 * = 'N': the Jacobi rotations are not accumulated.
65 *
66 * M (input) INTEGER
67 * The number of rows of the input matrix A. M >= 0.
68 *
69 * N (input) INTEGER
70 * The number of columns of the input matrix A.
71 * M >= N >= 0.
72 *
73 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
74 * On entry, M-by-N matrix A, such that A*diag(D) represents
75 * the input matrix.
76 * On exit,
77 * A_onexit * D_onexit represents the input matrix A*diag(D)
78 * post-multiplied by a sequence of Jacobi rotations, where the
79 * rotation threshold and the total number of sweeps are given in
80 * TOL and NSWEEP, respectively.
81 * (See the descriptions of D, TOL and NSWEEP.)
82 *
83 * LDA (input) INTEGER
84 * The leading dimension of the array A. LDA >= max(1,M).
85 *
86 * D (input/workspace/output) DOUBLE PRECISION array, dimension (N)
87 * The array D accumulates the scaling factors from the fast scaled
88 * Jacobi rotations.
89 * On entry, A*diag(D) represents the input matrix.
90 * On exit, A_onexit*diag(D_onexit) represents the input matrix
91 * post-multiplied by a sequence of Jacobi rotations, where the
92 * rotation threshold and the total number of sweeps are given in
93 * TOL and NSWEEP, respectively.
94 * (See the descriptions of A, TOL and NSWEEP.)
95 *
96 * SVA (input/workspace/output) DOUBLE PRECISION array, dimension (N)
97 * On entry, SVA contains the Euclidean norms of the columns of
98 * the matrix A*diag(D).
99 * On exit, SVA contains the Euclidean norms of the columns of
100 * the matrix onexit*diag(D_onexit).
101 *
102 * MV (input) INTEGER
103 * If JOBV .EQ. 'A', then MV rows of V are post-multipled by a
104 * sequence of Jacobi rotations.
105 * If JOBV = 'N', then MV is not referenced.
106 *
107 * V (input/output) DOUBLE PRECISION array, dimension (LDV,N)
108 * If JOBV .EQ. 'V' then N rows of V are post-multipled by a
109 * sequence of Jacobi rotations.
110 * If JOBV .EQ. 'A' then MV rows of V are post-multipled by a
111 * sequence of Jacobi rotations.
112 * If JOBV = 'N', then V is not referenced.
113 *
114 * LDV (input) INTEGER
115 * The leading dimension of the array V, LDV >= 1.
116 * If JOBV = 'V', LDV .GE. N.
117 * If JOBV = 'A', LDV .GE. MV.
118 *
119 * EPS (input) DOUBLE PRECISION
120 * EPS = DLAMCH('Epsilon')
121 *
122 * SFMIN (input) DOUBLE PRECISION
123 * SFMIN = DLAMCH('Safe Minimum')
124 *
125 * TOL (input) DOUBLE PRECISION
126 * TOL is the threshold for Jacobi rotations. For a pair
127 * A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
128 * applied only if DABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.
129 *
130 * NSWEEP (input) INTEGER
131 * NSWEEP is the number of sweeps of Jacobi rotations to be
132 * performed.
133 *
134 * WORK (workspace) DOUBLE PRECISION array, dimension (LWORK)
135 *
136 * LWORK (input) INTEGER
137 * LWORK is the dimension of WORK. LWORK .GE. M.
138 *
139 * INFO (output) INTEGER
140 * = 0 : successful exit.
141 * < 0 : if INFO = -i, then the i-th argument had an illegal value
142 *
143 * =====================================================================
144 *
145 * .. Local Parameters ..
146 DOUBLE PRECISION ZERO, HALF, ONE, TWO
147 PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,
148 $ TWO = 2.0D0 )
149 * ..
150 * .. Local Scalars ..
151 DOUBLE PRECISION AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
152 $ BIGTHETA, CS, MXAAPQ, MXSINJ, ROOTBIG, ROOTEPS,
153 $ ROOTSFMIN, ROOTTOL, SMALL, SN, T, TEMP1, THETA,
154 $ THSIGN
155 INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1,
156 $ ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, NBL,
157 $ NOTROT, p, PSKIPPED, q, ROWSKIP, SWBAND
158 LOGICAL APPLV, ROTOK, RSVEC
159 * ..
160 * .. Local Arrays ..
161 DOUBLE PRECISION FASTR( 5 )
162 * ..
163 * .. Intrinsic Functions ..
164 INTRINSIC DABS, DMAX1, DBLE, MIN0, DSIGN, DSQRT
165 * ..
166 * .. External Functions ..
167 DOUBLE PRECISION DDOT, DNRM2
168 INTEGER IDAMAX
169 LOGICAL LSAME
170 EXTERNAL IDAMAX, LSAME, DDOT, DNRM2
171 * ..
172 * .. External Subroutines ..
173 EXTERNAL DAXPY, DCOPY, DLASCL, DLASSQ, DROTM, DSWAP
174 * ..
175 * .. Executable Statements ..
176 *
177 * Test the input parameters.
178 *
179 APPLV = LSAME( JOBV, 'A' )
180 RSVEC = LSAME( JOBV, 'V' )
181 IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN
182 INFO = -1
183 ELSE IF( M.LT.0 ) THEN
184 INFO = -2
185 ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
186 INFO = -3
187 ELSE IF( LDA.LT.M ) THEN
188 INFO = -5
189 ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN
190 INFO = -8
191 ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR.
192 $ ( APPLV.AND.( LDV.LT.MV ) ) ) THEN
193 INFO = -10
194 ELSE IF( TOL.LE.EPS ) THEN
195 INFO = -13
196 ELSE IF( NSWEEP.LT.0 ) THEN
197 INFO = -14
198 ELSE IF( LWORK.LT.M ) THEN
199 INFO = -16
200 ELSE
201 INFO = 0
202 END IF
203 *
204 * #:(
205 IF( INFO.NE.0 ) THEN
206 CALL XERBLA( 'DGSVJ0', -INFO )
207 RETURN
208 END IF
209 *
210 IF( RSVEC ) THEN
211 MVL = N
212 ELSE IF( APPLV ) THEN
213 MVL = MV
214 END IF
215 RSVEC = RSVEC .OR. APPLV
216
217 ROOTEPS = DSQRT( EPS )
218 ROOTSFMIN = DSQRT( SFMIN )
219 SMALL = SFMIN / EPS
220 BIG = ONE / SFMIN
221 ROOTBIG = ONE / ROOTSFMIN
222 BIGTHETA = ONE / ROOTEPS
223 ROOTTOL = DSQRT( TOL )
224 *
225 * -#- Row-cyclic Jacobi SVD algorithm with column pivoting -#-
226 *
227 EMPTSW = ( N*( N-1 ) ) / 2
228 NOTROT = 0
229 FASTR( 1 ) = ZERO
230 *
231 * -#- Row-cyclic pivot strategy with de Rijk's pivoting -#-
232 *
233
234 SWBAND = 0
235 *[TP] SWBAND is a tuning parameter. It is meaningful and effective
236 * if SGESVJ is used as a computational routine in the preconditioned
237 * Jacobi SVD algorithm SGESVJ. For sweeps i=1:SWBAND the procedure
238 * ......
239
240 KBL = MIN0( 8, N )
241 *[TP] KBL is a tuning parameter that defines the tile size in the
242 * tiling of the p-q loops of pivot pairs. In general, an optimal
243 * value of KBL depends on the matrix dimensions and on the
244 * parameters of the computer's memory.
245 *
246 NBL = N / KBL
247 IF( ( NBL*KBL ).NE.N )NBL = NBL + 1
248
249 BLSKIP = ( KBL**2 ) + 1
250 *[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
251
252 ROWSKIP = MIN0( 5, KBL )
253 *[TP] ROWSKIP is a tuning parameter.
254
255 LKAHEAD = 1
256 *[TP] LKAHEAD is a tuning parameter.
257 SWBAND = 0
258 PSKIPPED = 0
259 *
260 DO 1993 i = 1, NSWEEP
261 * .. go go go ...
262 *
263 MXAAPQ = ZERO
264 MXSINJ = ZERO
265 ISWROT = 0
266 *
267 NOTROT = 0
268 PSKIPPED = 0
269 *
270 DO 2000 ibr = 1, NBL
271
272 igl = ( ibr-1 )*KBL + 1
273 *
274 DO 1002 ir1 = 0, MIN0( LKAHEAD, NBL-ibr )
275 *
276 igl = igl + ir1*KBL
277 *
278 DO 2001 p = igl, MIN0( igl+KBL-1, N-1 )
279
280 * .. de Rijk's pivoting
281 q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
282 IF( p.NE.q ) THEN
283 CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
284 IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1,
285 $ V( 1, q ), 1 )
286 TEMP1 = SVA( p )
287 SVA( p ) = SVA( q )
288 SVA( q ) = TEMP1
289 TEMP1 = D( p )
290 D( p ) = D( q )
291 D( q ) = TEMP1
292 END IF
293 *
294 IF( ir1.EQ.0 ) THEN
295 *
296 * Column norms are periodically updated by explicit
297 * norm computation.
298 * Caveat:
299 * Some BLAS implementations compute DNRM2(M,A(1,p),1)
300 * as DSQRT(DDOT(M,A(1,p),1,A(1,p),1)), which may result in
301 * overflow for ||A(:,p)||_2 > DSQRT(overflow_threshold), and
302 * undeflow for ||A(:,p)||_2 < DSQRT(underflow_threshold).
303 * Hence, DNRM2 cannot be trusted, not even in the case when
304 * the true norm is far from the under(over)flow boundaries.
305 * If properly implemented DNRM2 is available, the IF-THEN-ELSE
306 * below should read "AAPP = DNRM2( M, A(1,p), 1 ) * D(p)".
307 *
308 IF( ( SVA( p ).LT.ROOTBIG ) .AND.
309 $ ( SVA( p ).GT.ROOTSFMIN ) ) THEN
310 SVA( p ) = DNRM2( M, A( 1, p ), 1 )*D( p )
311 ELSE
312 TEMP1 = ZERO
313 AAPP = ONE
314 CALL DLASSQ( M, A( 1, p ), 1, TEMP1, AAPP )
315 SVA( p ) = TEMP1*DSQRT( AAPP )*D( p )
316 END IF
317 AAPP = SVA( p )
318 ELSE
319 AAPP = SVA( p )
320 END IF
321
322 *
323 IF( AAPP.GT.ZERO ) THEN
324 *
325 PSKIPPED = 0
326 *
327 DO 2002 q = p + 1, MIN0( igl+KBL-1, N )
328 *
329 AAQQ = SVA( q )
330
331 IF( AAQQ.GT.ZERO ) THEN
332 *
333 AAPP0 = AAPP
334 IF( AAQQ.GE.ONE ) THEN
335 ROTOK = ( SMALL*AAPP ).LE.AAQQ
336 IF( AAPP.LT.( BIG / AAQQ ) ) THEN
337 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
338 $ q ), 1 )*D( p )*D( q ) / AAQQ )
339 $ / AAPP
340 ELSE
341 CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
342 CALL DLASCL( 'G', 0, 0, AAPP, D( p ),
343 $ M, 1, WORK, LDA, IERR )
344 AAPQ = DDOT( M, WORK, 1, A( 1, q ),
345 $ 1 )*D( q ) / AAQQ
346 END IF
347 ELSE
348 ROTOK = AAPP.LE.( AAQQ / SMALL )
349 IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
350 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
351 $ q ), 1 )*D( p )*D( q ) / AAQQ )
352 $ / AAPP
353 ELSE
354 CALL DCOPY( M, A( 1, q ), 1, WORK, 1 )
355 CALL DLASCL( 'G', 0, 0, AAQQ, D( q ),
356 $ M, 1, WORK, LDA, IERR )
357 AAPQ = DDOT( M, WORK, 1, A( 1, p ),
358 $ 1 )*D( p ) / AAPP
359 END IF
360 END IF
361 *
362 MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) )
363 *
364 * TO rotate or NOT to rotate, THAT is the question ...
365 *
366 IF( DABS( AAPQ ).GT.TOL ) THEN
367 *
368 * .. rotate
369 * ROTATED = ROTATED + ONE
370 *
371 IF( ir1.EQ.0 ) THEN
372 NOTROT = 0
373 PSKIPPED = 0
374 ISWROT = ISWROT + 1
375 END IF
376 *
377 IF( ROTOK ) THEN
378 *
379 AQOAP = AAQQ / AAPP
380 APOAQ = AAPP / AAQQ
381 THETA = -HALF*DABS( AQOAP-APOAQ )/AAPQ
382 *
383 IF( DABS( THETA ).GT.BIGTHETA ) THEN
384 *
385 T = HALF / THETA
386 FASTR( 3 ) = T*D( p ) / D( q )
387 FASTR( 4 ) = -T*D( q ) / D( p )
388 CALL DROTM( M, A( 1, p ), 1,
389 $ A( 1, q ), 1, FASTR )
390 IF( RSVEC )CALL DROTM( MVL,
391 $ V( 1, p ), 1,
392 $ V( 1, q ), 1,
393 $ FASTR )
394 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
395 $ ONE+T*APOAQ*AAPQ ) )
396 AAPP = AAPP*DSQRT( DMAX1( ZERO,
397 $ ONE-T*AQOAP*AAPQ ) )
398 MXSINJ = DMAX1( MXSINJ, DABS( T ) )
399 *
400 ELSE
401 *
402 * .. choose correct signum for THETA and rotate
403 *
404 THSIGN = -DSIGN( ONE, AAPQ )
405 T = ONE / ( THETA+THSIGN*
406 $ DSQRT( ONE+THETA*THETA ) )
407 CS = DSQRT( ONE / ( ONE+T*T ) )
408 SN = T*CS
409 *
410 MXSINJ = DMAX1( MXSINJ, DABS( SN ) )
411 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
412 $ ONE+T*APOAQ*AAPQ ) )
413 AAPP = AAPP*DSQRT( DMAX1( ZERO,
414 $ ONE-T*AQOAP*AAPQ ) )
415 *
416 APOAQ = D( p ) / D( q )
417 AQOAP = D( q ) / D( p )
418 IF( D( p ).GE.ONE ) THEN
419 IF( D( q ).GE.ONE ) THEN
420 FASTR( 3 ) = T*APOAQ
421 FASTR( 4 ) = -T*AQOAP
422 D( p ) = D( p )*CS
423 D( q ) = D( q )*CS
424 CALL DROTM( M, A( 1, p ), 1,
425 $ A( 1, q ), 1,
426 $ FASTR )
427 IF( RSVEC )CALL DROTM( MVL,
428 $ V( 1, p ), 1, V( 1, q ),
429 $ 1, FASTR )
430 ELSE
431 CALL DAXPY( M, -T*AQOAP,
432 $ A( 1, q ), 1,
433 $ A( 1, p ), 1 )
434 CALL DAXPY( M, CS*SN*APOAQ,
435 $ A( 1, p ), 1,
436 $ A( 1, q ), 1 )
437 D( p ) = D( p )*CS
438 D( q ) = D( q ) / CS
439 IF( RSVEC ) THEN
440 CALL DAXPY( MVL, -T*AQOAP,
441 $ V( 1, q ), 1,
442 $ V( 1, p ), 1 )
443 CALL DAXPY( MVL,
444 $ CS*SN*APOAQ,
445 $ V( 1, p ), 1,
446 $ V( 1, q ), 1 )
447 END IF
448 END IF
449 ELSE
450 IF( D( q ).GE.ONE ) THEN
451 CALL DAXPY( M, T*APOAQ,
452 $ A( 1, p ), 1,
453 $ A( 1, q ), 1 )
454 CALL DAXPY( M, -CS*SN*AQOAP,
455 $ A( 1, q ), 1,
456 $ A( 1, p ), 1 )
457 D( p ) = D( p ) / CS
458 D( q ) = D( q )*CS
459 IF( RSVEC ) THEN
460 CALL DAXPY( MVL, T*APOAQ,
461 $ V( 1, p ), 1,
462 $ V( 1, q ), 1 )
463 CALL DAXPY( MVL,
464 $ -CS*SN*AQOAP,
465 $ V( 1, q ), 1,
466 $ V( 1, p ), 1 )
467 END IF
468 ELSE
469 IF( D( p ).GE.D( q ) ) THEN
470 CALL DAXPY( M, -T*AQOAP,
471 $ A( 1, q ), 1,
472 $ A( 1, p ), 1 )
473 CALL DAXPY( M, CS*SN*APOAQ,
474 $ A( 1, p ), 1,
475 $ A( 1, q ), 1 )
476 D( p ) = D( p )*CS
477 D( q ) = D( q ) / CS
478 IF( RSVEC ) THEN
479 CALL DAXPY( MVL,
480 $ -T*AQOAP,
481 $ V( 1, q ), 1,
482 $ V( 1, p ), 1 )
483 CALL DAXPY( MVL,
484 $ CS*SN*APOAQ,
485 $ V( 1, p ), 1,
486 $ V( 1, q ), 1 )
487 END IF
488 ELSE
489 CALL DAXPY( M, T*APOAQ,
490 $ A( 1, p ), 1,
491 $ A( 1, q ), 1 )
492 CALL DAXPY( M,
493 $ -CS*SN*AQOAP,
494 $ A( 1, q ), 1,
495 $ A( 1, p ), 1 )
496 D( p ) = D( p ) / CS
497 D( q ) = D( q )*CS
498 IF( RSVEC ) THEN
499 CALL DAXPY( MVL,
500 $ T*APOAQ, V( 1, p ),
501 $ 1, V( 1, q ), 1 )
502 CALL DAXPY( MVL,
503 $ -CS*SN*AQOAP,
504 $ V( 1, q ), 1,
505 $ V( 1, p ), 1 )
506 END IF
507 END IF
508 END IF
509 END IF
510 END IF
511 *
512 ELSE
513 * .. have to use modified Gram-Schmidt like transformation
514 CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
515 CALL DLASCL( 'G', 0, 0, AAPP, ONE, M,
516 $ 1, WORK, LDA, IERR )
517 CALL DLASCL( 'G', 0, 0, AAQQ, ONE, M,
518 $ 1, A( 1, q ), LDA, IERR )
519 TEMP1 = -AAPQ*D( p ) / D( q )
520 CALL DAXPY( M, TEMP1, WORK, 1,
521 $ A( 1, q ), 1 )
522 CALL DLASCL( 'G', 0, 0, ONE, AAQQ, M,
523 $ 1, A( 1, q ), LDA, IERR )
524 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
525 $ ONE-AAPQ*AAPQ ) )
526 MXSINJ = DMAX1( MXSINJ, SFMIN )
527 END IF
528 * END IF ROTOK THEN ... ELSE
529 *
530 * In the case of cancellation in updating SVA(q), SVA(p)
531 * recompute SVA(q), SVA(p).
532 IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
533 $ THEN
534 IF( ( AAQQ.LT.ROOTBIG ) .AND.
535 $ ( AAQQ.GT.ROOTSFMIN ) ) THEN
536 SVA( q ) = DNRM2( M, A( 1, q ), 1 )*
537 $ D( q )
538 ELSE
539 T = ZERO
540 AAQQ = ONE
541 CALL DLASSQ( M, A( 1, q ), 1, T,
542 $ AAQQ )
543 SVA( q ) = T*DSQRT( AAQQ )*D( q )
544 END IF
545 END IF
546 IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN
547 IF( ( AAPP.LT.ROOTBIG ) .AND.
548 $ ( AAPP.GT.ROOTSFMIN ) ) THEN
549 AAPP = DNRM2( M, A( 1, p ), 1 )*
550 $ D( p )
551 ELSE
552 T = ZERO
553 AAPP = ONE
554 CALL DLASSQ( M, A( 1, p ), 1, T,
555 $ AAPP )
556 AAPP = T*DSQRT( AAPP )*D( p )
557 END IF
558 SVA( p ) = AAPP
559 END IF
560 *
561 ELSE
562 * A(:,p) and A(:,q) already numerically orthogonal
563 IF( ir1.EQ.0 )NOTROT = NOTROT + 1
564 PSKIPPED = PSKIPPED + 1
565 END IF
566 ELSE
567 * A(:,q) is zero column
568 IF( ir1.EQ.0 )NOTROT = NOTROT + 1
569 PSKIPPED = PSKIPPED + 1
570 END IF
571 *
572 IF( ( i.LE.SWBAND ) .AND.
573 $ ( PSKIPPED.GT.ROWSKIP ) ) THEN
574 IF( ir1.EQ.0 )AAPP = -AAPP
575 NOTROT = 0
576 GO TO 2103
577 END IF
578 *
579 2002 CONTINUE
580 * END q-LOOP
581 *
582 2103 CONTINUE
583 * bailed out of q-loop
584
585 SVA( p ) = AAPP
586
587 ELSE
588 SVA( p ) = AAPP
589 IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) )
590 $ NOTROT = NOTROT + MIN0( igl+KBL-1, N ) - p
591 END IF
592 *
593 2001 CONTINUE
594 * end of the p-loop
595 * end of doing the block ( ibr, ibr )
596 1002 CONTINUE
597 * end of ir1-loop
598 *
599 *........................................................
600 * ... go to the off diagonal blocks
601 *
602 igl = ( ibr-1 )*KBL + 1
603 *
604 DO 2010 jbc = ibr + 1, NBL
605 *
606 jgl = ( jbc-1 )*KBL + 1
607 *
608 * doing the block at ( ibr, jbc )
609 *
610 IJBLSK = 0
611 DO 2100 p = igl, MIN0( igl+KBL-1, N )
612 *
613 AAPP = SVA( p )
614 *
615 IF( AAPP.GT.ZERO ) THEN
616 *
617 PSKIPPED = 0
618 *
619 DO 2200 q = jgl, MIN0( jgl+KBL-1, N )
620 *
621 AAQQ = SVA( q )
622 *
623 IF( AAQQ.GT.ZERO ) THEN
624 AAPP0 = AAPP
625 *
626 * -#- M x 2 Jacobi SVD -#-
627 *
628 * -#- Safe Gram matrix computation -#-
629 *
630 IF( AAQQ.GE.ONE ) THEN
631 IF( AAPP.GE.AAQQ ) THEN
632 ROTOK = ( SMALL*AAPP ).LE.AAQQ
633 ELSE
634 ROTOK = ( SMALL*AAQQ ).LE.AAPP
635 END IF
636 IF( AAPP.LT.( BIG / AAQQ ) ) THEN
637 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
638 $ q ), 1 )*D( p )*D( q ) / AAQQ )
639 $ / AAPP
640 ELSE
641 CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
642 CALL DLASCL( 'G', 0, 0, AAPP, D( p ),
643 $ M, 1, WORK, LDA, IERR )
644 AAPQ = DDOT( M, WORK, 1, A( 1, q ),
645 $ 1 )*D( q ) / AAQQ
646 END IF
647 ELSE
648 IF( AAPP.GE.AAQQ ) THEN
649 ROTOK = AAPP.LE.( AAQQ / SMALL )
650 ELSE
651 ROTOK = AAQQ.LE.( AAPP / SMALL )
652 END IF
653 IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
654 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
655 $ q ), 1 )*D( p )*D( q ) / AAQQ )
656 $ / AAPP
657 ELSE
658 CALL DCOPY( M, A( 1, q ), 1, WORK, 1 )
659 CALL DLASCL( 'G', 0, 0, AAQQ, D( q ),
660 $ M, 1, WORK, LDA, IERR )
661 AAPQ = DDOT( M, WORK, 1, A( 1, p ),
662 $ 1 )*D( p ) / AAPP
663 END IF
664 END IF
665 *
666 MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) )
667 *
668 * TO rotate or NOT to rotate, THAT is the question ...
669 *
670 IF( DABS( AAPQ ).GT.TOL ) THEN
671 NOTROT = 0
672 * ROTATED = ROTATED + 1
673 PSKIPPED = 0
674 ISWROT = ISWROT + 1
675 *
676 IF( ROTOK ) THEN
677 *
678 AQOAP = AAQQ / AAPP
679 APOAQ = AAPP / AAQQ
680 THETA = -HALF*DABS( AQOAP-APOAQ )/AAPQ
681 IF( AAQQ.GT.AAPP0 )THETA = -THETA
682 *
683 IF( DABS( THETA ).GT.BIGTHETA ) THEN
684 T = HALF / THETA
685 FASTR( 3 ) = T*D( p ) / D( q )
686 FASTR( 4 ) = -T*D( q ) / D( p )
687 CALL DROTM( M, A( 1, p ), 1,
688 $ A( 1, q ), 1, FASTR )
689 IF( RSVEC )CALL DROTM( MVL,
690 $ V( 1, p ), 1,
691 $ V( 1, q ), 1,
692 $ FASTR )
693 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
694 $ ONE+T*APOAQ*AAPQ ) )
695 AAPP = AAPP*DSQRT( DMAX1( ZERO,
696 $ ONE-T*AQOAP*AAPQ ) )
697 MXSINJ = DMAX1( MXSINJ, DABS( T ) )
698 ELSE
699 *
700 * .. choose correct signum for THETA and rotate
701 *
702 THSIGN = -DSIGN( ONE, AAPQ )
703 IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
704 T = ONE / ( THETA+THSIGN*
705 $ DSQRT( ONE+THETA*THETA ) )
706 CS = DSQRT( ONE / ( ONE+T*T ) )
707 SN = T*CS
708 MXSINJ = DMAX1( MXSINJ, DABS( SN ) )
709 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
710 $ ONE+T*APOAQ*AAPQ ) )
711 AAPP = AAPP*DSQRT( DMAX1( ZERO,
712 $ ONE-T*AQOAP*AAPQ ) )
713 *
714 APOAQ = D( p ) / D( q )
715 AQOAP = D( q ) / D( p )
716 IF( D( p ).GE.ONE ) THEN
717 *
718 IF( D( q ).GE.ONE ) THEN
719 FASTR( 3 ) = T*APOAQ
720 FASTR( 4 ) = -T*AQOAP
721 D( p ) = D( p )*CS
722 D( q ) = D( q )*CS
723 CALL DROTM( M, A( 1, p ), 1,
724 $ A( 1, q ), 1,
725 $ FASTR )
726 IF( RSVEC )CALL DROTM( MVL,
727 $ V( 1, p ), 1, V( 1, q ),
728 $ 1, FASTR )
729 ELSE
730 CALL DAXPY( M, -T*AQOAP,
731 $ A( 1, q ), 1,
732 $ A( 1, p ), 1 )
733 CALL DAXPY( M, CS*SN*APOAQ,
734 $ A( 1, p ), 1,
735 $ A( 1, q ), 1 )
736 IF( RSVEC ) THEN
737 CALL DAXPY( MVL, -T*AQOAP,
738 $ V( 1, q ), 1,
739 $ V( 1, p ), 1 )
740 CALL DAXPY( MVL,
741 $ CS*SN*APOAQ,
742 $ V( 1, p ), 1,
743 $ V( 1, q ), 1 )
744 END IF
745 D( p ) = D( p )*CS
746 D( q ) = D( q ) / CS
747 END IF
748 ELSE
749 IF( D( q ).GE.ONE ) THEN
750 CALL DAXPY( M, T*APOAQ,
751 $ A( 1, p ), 1,
752 $ A( 1, q ), 1 )
753 CALL DAXPY( M, -CS*SN*AQOAP,
754 $ A( 1, q ), 1,
755 $ A( 1, p ), 1 )
756 IF( RSVEC ) THEN
757 CALL DAXPY( MVL, T*APOAQ,
758 $ V( 1, p ), 1,
759 $ V( 1, q ), 1 )
760 CALL DAXPY( MVL,
761 $ -CS*SN*AQOAP,
762 $ V( 1, q ), 1,
763 $ V( 1, p ), 1 )
764 END IF
765 D( p ) = D( p ) / CS
766 D( q ) = D( q )*CS
767 ELSE
768 IF( D( p ).GE.D( q ) ) THEN
769 CALL DAXPY( M, -T*AQOAP,
770 $ A( 1, q ), 1,
771 $ A( 1, p ), 1 )
772 CALL DAXPY( M, CS*SN*APOAQ,
773 $ A( 1, p ), 1,
774 $ A( 1, q ), 1 )
775 D( p ) = D( p )*CS
776 D( q ) = D( q ) / CS
777 IF( RSVEC ) THEN
778 CALL DAXPY( MVL,
779 $ -T*AQOAP,
780 $ V( 1, q ), 1,
781 $ V( 1, p ), 1 )
782 CALL DAXPY( MVL,
783 $ CS*SN*APOAQ,
784 $ V( 1, p ), 1,
785 $ V( 1, q ), 1 )
786 END IF
787 ELSE
788 CALL DAXPY( M, T*APOAQ,
789 $ A( 1, p ), 1,
790 $ A( 1, q ), 1 )
791 CALL DAXPY( M,
792 $ -CS*SN*AQOAP,
793 $ A( 1, q ), 1,
794 $ A( 1, p ), 1 )
795 D( p ) = D( p ) / CS
796 D( q ) = D( q )*CS
797 IF( RSVEC ) THEN
798 CALL DAXPY( MVL,
799 $ T*APOAQ, V( 1, p ),
800 $ 1, V( 1, q ), 1 )
801 CALL DAXPY( MVL,
802 $ -CS*SN*AQOAP,
803 $ V( 1, q ), 1,
804 $ V( 1, p ), 1 )
805 END IF
806 END IF
807 END IF
808 END IF
809 END IF
810 *
811 ELSE
812 IF( AAPP.GT.AAQQ ) THEN
813 CALL DCOPY( M, A( 1, p ), 1, WORK,
814 $ 1 )
815 CALL DLASCL( 'G', 0, 0, AAPP, ONE,
816 $ M, 1, WORK, LDA, IERR )
817 CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
818 $ M, 1, A( 1, q ), LDA,
819 $ IERR )
820 TEMP1 = -AAPQ*D( p ) / D( q )
821 CALL DAXPY( M, TEMP1, WORK, 1,
822 $ A( 1, q ), 1 )
823 CALL DLASCL( 'G', 0, 0, ONE, AAQQ,
824 $ M, 1, A( 1, q ), LDA,
825 $ IERR )
826 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
827 $ ONE-AAPQ*AAPQ ) )
828 MXSINJ = DMAX1( MXSINJ, SFMIN )
829 ELSE
830 CALL DCOPY( M, A( 1, q ), 1, WORK,
831 $ 1 )
832 CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
833 $ M, 1, WORK, LDA, IERR )
834 CALL DLASCL( 'G', 0, 0, AAPP, ONE,
835 $ M, 1, A( 1, p ), LDA,
836 $ IERR )
837 TEMP1 = -AAPQ*D( q ) / D( p )
838 CALL DAXPY( M, TEMP1, WORK, 1,
839 $ A( 1, p ), 1 )
840 CALL DLASCL( 'G', 0, 0, ONE, AAPP,
841 $ M, 1, A( 1, p ), LDA,
842 $ IERR )
843 SVA( p ) = AAPP*DSQRT( DMAX1( ZERO,
844 $ ONE-AAPQ*AAPQ ) )
845 MXSINJ = DMAX1( MXSINJ, SFMIN )
846 END IF
847 END IF
848 * END IF ROTOK THEN ... ELSE
849 *
850 * In the case of cancellation in updating SVA(q)
851 * .. recompute SVA(q)
852 IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
853 $ THEN
854 IF( ( AAQQ.LT.ROOTBIG ) .AND.
855 $ ( AAQQ.GT.ROOTSFMIN ) ) THEN
856 SVA( q ) = DNRM2( M, A( 1, q ), 1 )*
857 $ D( q )
858 ELSE
859 T = ZERO
860 AAQQ = ONE
861 CALL DLASSQ( M, A( 1, q ), 1, T,
862 $ AAQQ )
863 SVA( q ) = T*DSQRT( AAQQ )*D( q )
864 END IF
865 END IF
866 IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
867 IF( ( AAPP.LT.ROOTBIG ) .AND.
868 $ ( AAPP.GT.ROOTSFMIN ) ) THEN
869 AAPP = DNRM2( M, A( 1, p ), 1 )*
870 $ D( p )
871 ELSE
872 T = ZERO
873 AAPP = ONE
874 CALL DLASSQ( M, A( 1, p ), 1, T,
875 $ AAPP )
876 AAPP = T*DSQRT( AAPP )*D( p )
877 END IF
878 SVA( p ) = AAPP
879 END IF
880 * end of OK rotation
881 ELSE
882 NOTROT = NOTROT + 1
883 PSKIPPED = PSKIPPED + 1
884 IJBLSK = IJBLSK + 1
885 END IF
886 ELSE
887 NOTROT = NOTROT + 1
888 PSKIPPED = PSKIPPED + 1
889 IJBLSK = IJBLSK + 1
890 END IF
891 *
892 IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) )
893 $ THEN
894 SVA( p ) = AAPP
895 NOTROT = 0
896 GO TO 2011
897 END IF
898 IF( ( i.LE.SWBAND ) .AND.
899 $ ( PSKIPPED.GT.ROWSKIP ) ) THEN
900 AAPP = -AAPP
901 NOTROT = 0
902 GO TO 2203
903 END IF
904 *
905 2200 CONTINUE
906 * end of the q-loop
907 2203 CONTINUE
908 *
909 SVA( p ) = AAPP
910 *
911 ELSE
912 IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
913 $ MIN0( jgl+KBL-1, N ) - jgl + 1
914 IF( AAPP.LT.ZERO )NOTROT = 0
915 END IF
916
917 2100 CONTINUE
918 * end of the p-loop
919 2010 CONTINUE
920 * end of the jbc-loop
921 2011 CONTINUE
922 *2011 bailed out of the jbc-loop
923 DO 2012 p = igl, MIN0( igl+KBL-1, N )
924 SVA( p ) = DABS( SVA( p ) )
925 2012 CONTINUE
926 *
927 2000 CONTINUE
928 *2000 :: end of the ibr-loop
929 *
930 * .. update SVA(N)
931 IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) )
932 $ THEN
933 SVA( N ) = DNRM2( M, A( 1, N ), 1 )*D( N )
934 ELSE
935 T = ZERO
936 AAPP = ONE
937 CALL DLASSQ( M, A( 1, N ), 1, T, AAPP )
938 SVA( N ) = T*DSQRT( AAPP )*D( N )
939 END IF
940 *
941 * Additional steering devices
942 *
943 IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR.
944 $ ( ISWROT.LE.N ) ) )SWBAND = i
945 *
946 IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.DBLE( N )*TOL ) .AND.
947 $ ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN
948 GO TO 1994
949 END IF
950 *
951 IF( NOTROT.GE.EMPTSW )GO TO 1994
952
953 1993 CONTINUE
954 * end i=1:NSWEEP loop
955 * #:) Reaching this point means that the procedure has comleted the given
956 * number of iterations.
957 INFO = NSWEEP - 1
958 GO TO 1995
959 1994 CONTINUE
960 * #:) Reaching this point means that during the i-th sweep all pivots were
961 * below the given tolerance, causing early exit.
962 *
963 INFO = 0
964 * #:) INFO = 0 confirms successful iterations.
965 1995 CONTINUE
966 *
967 * Sort the vector D.
968 DO 5991 p = 1, N - 1
969 q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
970 IF( p.NE.q ) THEN
971 TEMP1 = SVA( p )
972 SVA( p ) = SVA( q )
973 SVA( q ) = TEMP1
974 TEMP1 = D( p )
975 D( p ) = D( q )
976 D( q ) = TEMP1
977 CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
978 IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
979 END IF
980 5991 CONTINUE
981 *
982 RETURN
983 * ..
984 * .. END OF DGSVJ0
985 * ..
986 END
2 $ 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 INTEGER INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP
22 DOUBLE PRECISION EPS, SFMIN, TOL
23 CHARACTER*1 JOBV
24 * ..
25 * .. Array Arguments ..
26 DOUBLE PRECISION A( LDA, * ), SVA( N ), D( N ), V( LDV, * ),
27 $ WORK( LWORK )
28 * ..
29 *
30 * Purpose
31 * =======
32 *
33 * DGSVJ0 is called from DGESVJ as a pre-processor and that is its main
34 * purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
35 * it does not check convergence (stopping criterion). Few tuning
36 * parameters (marked by [TP]) are available for the implementer.
37 *
38 * Further Details
39 * ~~~~~~~~~~~~~~~
40 * DGSVJ0 is used just to enable SGESVJ to call a simplified version of
41 * itself to work on a submatrix of the original matrix.
42 *
43 * Contributors
44 * ~~~~~~~~~~~~
45 * Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
46 *
47 * Bugs, Examples and Comments
48 * ~~~~~~~~~~~~~~~~~~~~~~~~~~~
49 * Please report all bugs and send interesting test examples and comments to
50 * drmac@math.hr. Thank you.
51 *
52 * Arguments
53 * =========
54 *
55 * JOBV (input) CHARACTER*1
56 * Specifies whether the output from this procedure is used
57 * to compute the matrix V:
58 * = 'V': the product of the Jacobi rotations is accumulated
59 * by postmulyiplying the N-by-N array V.
60 * (See the description of V.)
61 * = 'A': the product of the Jacobi rotations is accumulated
62 * by postmulyiplying the MV-by-N array V.
63 * (See the descriptions of MV and V.)
64 * = 'N': the Jacobi rotations are not accumulated.
65 *
66 * M (input) INTEGER
67 * The number of rows of the input matrix A. M >= 0.
68 *
69 * N (input) INTEGER
70 * The number of columns of the input matrix A.
71 * M >= N >= 0.
72 *
73 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
74 * On entry, M-by-N matrix A, such that A*diag(D) represents
75 * the input matrix.
76 * On exit,
77 * A_onexit * D_onexit represents the input matrix A*diag(D)
78 * post-multiplied by a sequence of Jacobi rotations, where the
79 * rotation threshold and the total number of sweeps are given in
80 * TOL and NSWEEP, respectively.
81 * (See the descriptions of D, TOL and NSWEEP.)
82 *
83 * LDA (input) INTEGER
84 * The leading dimension of the array A. LDA >= max(1,M).
85 *
86 * D (input/workspace/output) DOUBLE PRECISION array, dimension (N)
87 * The array D accumulates the scaling factors from the fast scaled
88 * Jacobi rotations.
89 * On entry, A*diag(D) represents the input matrix.
90 * On exit, A_onexit*diag(D_onexit) represents the input matrix
91 * post-multiplied by a sequence of Jacobi rotations, where the
92 * rotation threshold and the total number of sweeps are given in
93 * TOL and NSWEEP, respectively.
94 * (See the descriptions of A, TOL and NSWEEP.)
95 *
96 * SVA (input/workspace/output) DOUBLE PRECISION array, dimension (N)
97 * On entry, SVA contains the Euclidean norms of the columns of
98 * the matrix A*diag(D).
99 * On exit, SVA contains the Euclidean norms of the columns of
100 * the matrix onexit*diag(D_onexit).
101 *
102 * MV (input) INTEGER
103 * If JOBV .EQ. 'A', then MV rows of V are post-multipled by a
104 * sequence of Jacobi rotations.
105 * If JOBV = 'N', then MV is not referenced.
106 *
107 * V (input/output) DOUBLE PRECISION array, dimension (LDV,N)
108 * If JOBV .EQ. 'V' then N rows of V are post-multipled by a
109 * sequence of Jacobi rotations.
110 * If JOBV .EQ. 'A' then MV rows of V are post-multipled by a
111 * sequence of Jacobi rotations.
112 * If JOBV = 'N', then V is not referenced.
113 *
114 * LDV (input) INTEGER
115 * The leading dimension of the array V, LDV >= 1.
116 * If JOBV = 'V', LDV .GE. N.
117 * If JOBV = 'A', LDV .GE. MV.
118 *
119 * EPS (input) DOUBLE PRECISION
120 * EPS = DLAMCH('Epsilon')
121 *
122 * SFMIN (input) DOUBLE PRECISION
123 * SFMIN = DLAMCH('Safe Minimum')
124 *
125 * TOL (input) DOUBLE PRECISION
126 * TOL is the threshold for Jacobi rotations. For a pair
127 * A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
128 * applied only if DABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.
129 *
130 * NSWEEP (input) INTEGER
131 * NSWEEP is the number of sweeps of Jacobi rotations to be
132 * performed.
133 *
134 * WORK (workspace) DOUBLE PRECISION array, dimension (LWORK)
135 *
136 * LWORK (input) INTEGER
137 * LWORK is the dimension of WORK. LWORK .GE. M.
138 *
139 * INFO (output) INTEGER
140 * = 0 : successful exit.
141 * < 0 : if INFO = -i, then the i-th argument had an illegal value
142 *
143 * =====================================================================
144 *
145 * .. Local Parameters ..
146 DOUBLE PRECISION ZERO, HALF, ONE, TWO
147 PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,
148 $ TWO = 2.0D0 )
149 * ..
150 * .. Local Scalars ..
151 DOUBLE PRECISION AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
152 $ BIGTHETA, CS, MXAAPQ, MXSINJ, ROOTBIG, ROOTEPS,
153 $ ROOTSFMIN, ROOTTOL, SMALL, SN, T, TEMP1, THETA,
154 $ THSIGN
155 INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1,
156 $ ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, NBL,
157 $ NOTROT, p, PSKIPPED, q, ROWSKIP, SWBAND
158 LOGICAL APPLV, ROTOK, RSVEC
159 * ..
160 * .. Local Arrays ..
161 DOUBLE PRECISION FASTR( 5 )
162 * ..
163 * .. Intrinsic Functions ..
164 INTRINSIC DABS, DMAX1, DBLE, MIN0, DSIGN, DSQRT
165 * ..
166 * .. External Functions ..
167 DOUBLE PRECISION DDOT, DNRM2
168 INTEGER IDAMAX
169 LOGICAL LSAME
170 EXTERNAL IDAMAX, LSAME, DDOT, DNRM2
171 * ..
172 * .. External Subroutines ..
173 EXTERNAL DAXPY, DCOPY, DLASCL, DLASSQ, DROTM, DSWAP
174 * ..
175 * .. Executable Statements ..
176 *
177 * Test the input parameters.
178 *
179 APPLV = LSAME( JOBV, 'A' )
180 RSVEC = LSAME( JOBV, 'V' )
181 IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN
182 INFO = -1
183 ELSE IF( M.LT.0 ) THEN
184 INFO = -2
185 ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
186 INFO = -3
187 ELSE IF( LDA.LT.M ) THEN
188 INFO = -5
189 ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN
190 INFO = -8
191 ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR.
192 $ ( APPLV.AND.( LDV.LT.MV ) ) ) THEN
193 INFO = -10
194 ELSE IF( TOL.LE.EPS ) THEN
195 INFO = -13
196 ELSE IF( NSWEEP.LT.0 ) THEN
197 INFO = -14
198 ELSE IF( LWORK.LT.M ) THEN
199 INFO = -16
200 ELSE
201 INFO = 0
202 END IF
203 *
204 * #:(
205 IF( INFO.NE.0 ) THEN
206 CALL XERBLA( 'DGSVJ0', -INFO )
207 RETURN
208 END IF
209 *
210 IF( RSVEC ) THEN
211 MVL = N
212 ELSE IF( APPLV ) THEN
213 MVL = MV
214 END IF
215 RSVEC = RSVEC .OR. APPLV
216
217 ROOTEPS = DSQRT( EPS )
218 ROOTSFMIN = DSQRT( SFMIN )
219 SMALL = SFMIN / EPS
220 BIG = ONE / SFMIN
221 ROOTBIG = ONE / ROOTSFMIN
222 BIGTHETA = ONE / ROOTEPS
223 ROOTTOL = DSQRT( TOL )
224 *
225 * -#- Row-cyclic Jacobi SVD algorithm with column pivoting -#-
226 *
227 EMPTSW = ( N*( N-1 ) ) / 2
228 NOTROT = 0
229 FASTR( 1 ) = ZERO
230 *
231 * -#- Row-cyclic pivot strategy with de Rijk's pivoting -#-
232 *
233
234 SWBAND = 0
235 *[TP] SWBAND is a tuning parameter. It is meaningful and effective
236 * if SGESVJ is used as a computational routine in the preconditioned
237 * Jacobi SVD algorithm SGESVJ. For sweeps i=1:SWBAND the procedure
238 * ......
239
240 KBL = MIN0( 8, N )
241 *[TP] KBL is a tuning parameter that defines the tile size in the
242 * tiling of the p-q loops of pivot pairs. In general, an optimal
243 * value of KBL depends on the matrix dimensions and on the
244 * parameters of the computer's memory.
245 *
246 NBL = N / KBL
247 IF( ( NBL*KBL ).NE.N )NBL = NBL + 1
248
249 BLSKIP = ( KBL**2 ) + 1
250 *[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
251
252 ROWSKIP = MIN0( 5, KBL )
253 *[TP] ROWSKIP is a tuning parameter.
254
255 LKAHEAD = 1
256 *[TP] LKAHEAD is a tuning parameter.
257 SWBAND = 0
258 PSKIPPED = 0
259 *
260 DO 1993 i = 1, NSWEEP
261 * .. go go go ...
262 *
263 MXAAPQ = ZERO
264 MXSINJ = ZERO
265 ISWROT = 0
266 *
267 NOTROT = 0
268 PSKIPPED = 0
269 *
270 DO 2000 ibr = 1, NBL
271
272 igl = ( ibr-1 )*KBL + 1
273 *
274 DO 1002 ir1 = 0, MIN0( LKAHEAD, NBL-ibr )
275 *
276 igl = igl + ir1*KBL
277 *
278 DO 2001 p = igl, MIN0( igl+KBL-1, N-1 )
279
280 * .. de Rijk's pivoting
281 q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
282 IF( p.NE.q ) THEN
283 CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
284 IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1,
285 $ V( 1, q ), 1 )
286 TEMP1 = SVA( p )
287 SVA( p ) = SVA( q )
288 SVA( q ) = TEMP1
289 TEMP1 = D( p )
290 D( p ) = D( q )
291 D( q ) = TEMP1
292 END IF
293 *
294 IF( ir1.EQ.0 ) THEN
295 *
296 * Column norms are periodically updated by explicit
297 * norm computation.
298 * Caveat:
299 * Some BLAS implementations compute DNRM2(M,A(1,p),1)
300 * as DSQRT(DDOT(M,A(1,p),1,A(1,p),1)), which may result in
301 * overflow for ||A(:,p)||_2 > DSQRT(overflow_threshold), and
302 * undeflow for ||A(:,p)||_2 < DSQRT(underflow_threshold).
303 * Hence, DNRM2 cannot be trusted, not even in the case when
304 * the true norm is far from the under(over)flow boundaries.
305 * If properly implemented DNRM2 is available, the IF-THEN-ELSE
306 * below should read "AAPP = DNRM2( M, A(1,p), 1 ) * D(p)".
307 *
308 IF( ( SVA( p ).LT.ROOTBIG ) .AND.
309 $ ( SVA( p ).GT.ROOTSFMIN ) ) THEN
310 SVA( p ) = DNRM2( M, A( 1, p ), 1 )*D( p )
311 ELSE
312 TEMP1 = ZERO
313 AAPP = ONE
314 CALL DLASSQ( M, A( 1, p ), 1, TEMP1, AAPP )
315 SVA( p ) = TEMP1*DSQRT( AAPP )*D( p )
316 END IF
317 AAPP = SVA( p )
318 ELSE
319 AAPP = SVA( p )
320 END IF
321
322 *
323 IF( AAPP.GT.ZERO ) THEN
324 *
325 PSKIPPED = 0
326 *
327 DO 2002 q = p + 1, MIN0( igl+KBL-1, N )
328 *
329 AAQQ = SVA( q )
330
331 IF( AAQQ.GT.ZERO ) THEN
332 *
333 AAPP0 = AAPP
334 IF( AAQQ.GE.ONE ) THEN
335 ROTOK = ( SMALL*AAPP ).LE.AAQQ
336 IF( AAPP.LT.( BIG / AAQQ ) ) THEN
337 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
338 $ q ), 1 )*D( p )*D( q ) / AAQQ )
339 $ / AAPP
340 ELSE
341 CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
342 CALL DLASCL( 'G', 0, 0, AAPP, D( p ),
343 $ M, 1, WORK, LDA, IERR )
344 AAPQ = DDOT( M, WORK, 1, A( 1, q ),
345 $ 1 )*D( q ) / AAQQ
346 END IF
347 ELSE
348 ROTOK = AAPP.LE.( AAQQ / SMALL )
349 IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
350 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
351 $ q ), 1 )*D( p )*D( q ) / AAQQ )
352 $ / AAPP
353 ELSE
354 CALL DCOPY( M, A( 1, q ), 1, WORK, 1 )
355 CALL DLASCL( 'G', 0, 0, AAQQ, D( q ),
356 $ M, 1, WORK, LDA, IERR )
357 AAPQ = DDOT( M, WORK, 1, A( 1, p ),
358 $ 1 )*D( p ) / AAPP
359 END IF
360 END IF
361 *
362 MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) )
363 *
364 * TO rotate or NOT to rotate, THAT is the question ...
365 *
366 IF( DABS( AAPQ ).GT.TOL ) THEN
367 *
368 * .. rotate
369 * ROTATED = ROTATED + ONE
370 *
371 IF( ir1.EQ.0 ) THEN
372 NOTROT = 0
373 PSKIPPED = 0
374 ISWROT = ISWROT + 1
375 END IF
376 *
377 IF( ROTOK ) THEN
378 *
379 AQOAP = AAQQ / AAPP
380 APOAQ = AAPP / AAQQ
381 THETA = -HALF*DABS( AQOAP-APOAQ )/AAPQ
382 *
383 IF( DABS( THETA ).GT.BIGTHETA ) THEN
384 *
385 T = HALF / THETA
386 FASTR( 3 ) = T*D( p ) / D( q )
387 FASTR( 4 ) = -T*D( q ) / D( p )
388 CALL DROTM( M, A( 1, p ), 1,
389 $ A( 1, q ), 1, FASTR )
390 IF( RSVEC )CALL DROTM( MVL,
391 $ V( 1, p ), 1,
392 $ V( 1, q ), 1,
393 $ FASTR )
394 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
395 $ ONE+T*APOAQ*AAPQ ) )
396 AAPP = AAPP*DSQRT( DMAX1( ZERO,
397 $ ONE-T*AQOAP*AAPQ ) )
398 MXSINJ = DMAX1( MXSINJ, DABS( T ) )
399 *
400 ELSE
401 *
402 * .. choose correct signum for THETA and rotate
403 *
404 THSIGN = -DSIGN( ONE, AAPQ )
405 T = ONE / ( THETA+THSIGN*
406 $ DSQRT( ONE+THETA*THETA ) )
407 CS = DSQRT( ONE / ( ONE+T*T ) )
408 SN = T*CS
409 *
410 MXSINJ = DMAX1( MXSINJ, DABS( SN ) )
411 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
412 $ ONE+T*APOAQ*AAPQ ) )
413 AAPP = AAPP*DSQRT( DMAX1( ZERO,
414 $ ONE-T*AQOAP*AAPQ ) )
415 *
416 APOAQ = D( p ) / D( q )
417 AQOAP = D( q ) / D( p )
418 IF( D( p ).GE.ONE ) THEN
419 IF( D( q ).GE.ONE ) THEN
420 FASTR( 3 ) = T*APOAQ
421 FASTR( 4 ) = -T*AQOAP
422 D( p ) = D( p )*CS
423 D( q ) = D( q )*CS
424 CALL DROTM( M, A( 1, p ), 1,
425 $ A( 1, q ), 1,
426 $ FASTR )
427 IF( RSVEC )CALL DROTM( MVL,
428 $ V( 1, p ), 1, V( 1, q ),
429 $ 1, FASTR )
430 ELSE
431 CALL DAXPY( M, -T*AQOAP,
432 $ A( 1, q ), 1,
433 $ A( 1, p ), 1 )
434 CALL DAXPY( M, CS*SN*APOAQ,
435 $ A( 1, p ), 1,
436 $ A( 1, q ), 1 )
437 D( p ) = D( p )*CS
438 D( q ) = D( q ) / CS
439 IF( RSVEC ) THEN
440 CALL DAXPY( MVL, -T*AQOAP,
441 $ V( 1, q ), 1,
442 $ V( 1, p ), 1 )
443 CALL DAXPY( MVL,
444 $ CS*SN*APOAQ,
445 $ V( 1, p ), 1,
446 $ V( 1, q ), 1 )
447 END IF
448 END IF
449 ELSE
450 IF( D( q ).GE.ONE ) THEN
451 CALL DAXPY( M, T*APOAQ,
452 $ A( 1, p ), 1,
453 $ A( 1, q ), 1 )
454 CALL DAXPY( M, -CS*SN*AQOAP,
455 $ A( 1, q ), 1,
456 $ A( 1, p ), 1 )
457 D( p ) = D( p ) / CS
458 D( q ) = D( q )*CS
459 IF( RSVEC ) THEN
460 CALL DAXPY( MVL, T*APOAQ,
461 $ V( 1, p ), 1,
462 $ V( 1, q ), 1 )
463 CALL DAXPY( MVL,
464 $ -CS*SN*AQOAP,
465 $ V( 1, q ), 1,
466 $ V( 1, p ), 1 )
467 END IF
468 ELSE
469 IF( D( p ).GE.D( q ) ) THEN
470 CALL DAXPY( M, -T*AQOAP,
471 $ A( 1, q ), 1,
472 $ A( 1, p ), 1 )
473 CALL DAXPY( M, CS*SN*APOAQ,
474 $ A( 1, p ), 1,
475 $ A( 1, q ), 1 )
476 D( p ) = D( p )*CS
477 D( q ) = D( q ) / CS
478 IF( RSVEC ) THEN
479 CALL DAXPY( MVL,
480 $ -T*AQOAP,
481 $ V( 1, q ), 1,
482 $ V( 1, p ), 1 )
483 CALL DAXPY( MVL,
484 $ CS*SN*APOAQ,
485 $ V( 1, p ), 1,
486 $ V( 1, q ), 1 )
487 END IF
488 ELSE
489 CALL DAXPY( M, T*APOAQ,
490 $ A( 1, p ), 1,
491 $ A( 1, q ), 1 )
492 CALL DAXPY( M,
493 $ -CS*SN*AQOAP,
494 $ A( 1, q ), 1,
495 $ A( 1, p ), 1 )
496 D( p ) = D( p ) / CS
497 D( q ) = D( q )*CS
498 IF( RSVEC ) THEN
499 CALL DAXPY( MVL,
500 $ T*APOAQ, V( 1, p ),
501 $ 1, V( 1, q ), 1 )
502 CALL DAXPY( MVL,
503 $ -CS*SN*AQOAP,
504 $ V( 1, q ), 1,
505 $ V( 1, p ), 1 )
506 END IF
507 END IF
508 END IF
509 END IF
510 END IF
511 *
512 ELSE
513 * .. have to use modified Gram-Schmidt like transformation
514 CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
515 CALL DLASCL( 'G', 0, 0, AAPP, ONE, M,
516 $ 1, WORK, LDA, IERR )
517 CALL DLASCL( 'G', 0, 0, AAQQ, ONE, M,
518 $ 1, A( 1, q ), LDA, IERR )
519 TEMP1 = -AAPQ*D( p ) / D( q )
520 CALL DAXPY( M, TEMP1, WORK, 1,
521 $ A( 1, q ), 1 )
522 CALL DLASCL( 'G', 0, 0, ONE, AAQQ, M,
523 $ 1, A( 1, q ), LDA, IERR )
524 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
525 $ ONE-AAPQ*AAPQ ) )
526 MXSINJ = DMAX1( MXSINJ, SFMIN )
527 END IF
528 * END IF ROTOK THEN ... ELSE
529 *
530 * In the case of cancellation in updating SVA(q), SVA(p)
531 * recompute SVA(q), SVA(p).
532 IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
533 $ THEN
534 IF( ( AAQQ.LT.ROOTBIG ) .AND.
535 $ ( AAQQ.GT.ROOTSFMIN ) ) THEN
536 SVA( q ) = DNRM2( M, A( 1, q ), 1 )*
537 $ D( q )
538 ELSE
539 T = ZERO
540 AAQQ = ONE
541 CALL DLASSQ( M, A( 1, q ), 1, T,
542 $ AAQQ )
543 SVA( q ) = T*DSQRT( AAQQ )*D( q )
544 END IF
545 END IF
546 IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN
547 IF( ( AAPP.LT.ROOTBIG ) .AND.
548 $ ( AAPP.GT.ROOTSFMIN ) ) THEN
549 AAPP = DNRM2( M, A( 1, p ), 1 )*
550 $ D( p )
551 ELSE
552 T = ZERO
553 AAPP = ONE
554 CALL DLASSQ( M, A( 1, p ), 1, T,
555 $ AAPP )
556 AAPP = T*DSQRT( AAPP )*D( p )
557 END IF
558 SVA( p ) = AAPP
559 END IF
560 *
561 ELSE
562 * A(:,p) and A(:,q) already numerically orthogonal
563 IF( ir1.EQ.0 )NOTROT = NOTROT + 1
564 PSKIPPED = PSKIPPED + 1
565 END IF
566 ELSE
567 * A(:,q) is zero column
568 IF( ir1.EQ.0 )NOTROT = NOTROT + 1
569 PSKIPPED = PSKIPPED + 1
570 END IF
571 *
572 IF( ( i.LE.SWBAND ) .AND.
573 $ ( PSKIPPED.GT.ROWSKIP ) ) THEN
574 IF( ir1.EQ.0 )AAPP = -AAPP
575 NOTROT = 0
576 GO TO 2103
577 END IF
578 *
579 2002 CONTINUE
580 * END q-LOOP
581 *
582 2103 CONTINUE
583 * bailed out of q-loop
584
585 SVA( p ) = AAPP
586
587 ELSE
588 SVA( p ) = AAPP
589 IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) )
590 $ NOTROT = NOTROT + MIN0( igl+KBL-1, N ) - p
591 END IF
592 *
593 2001 CONTINUE
594 * end of the p-loop
595 * end of doing the block ( ibr, ibr )
596 1002 CONTINUE
597 * end of ir1-loop
598 *
599 *........................................................
600 * ... go to the off diagonal blocks
601 *
602 igl = ( ibr-1 )*KBL + 1
603 *
604 DO 2010 jbc = ibr + 1, NBL
605 *
606 jgl = ( jbc-1 )*KBL + 1
607 *
608 * doing the block at ( ibr, jbc )
609 *
610 IJBLSK = 0
611 DO 2100 p = igl, MIN0( igl+KBL-1, N )
612 *
613 AAPP = SVA( p )
614 *
615 IF( AAPP.GT.ZERO ) THEN
616 *
617 PSKIPPED = 0
618 *
619 DO 2200 q = jgl, MIN0( jgl+KBL-1, N )
620 *
621 AAQQ = SVA( q )
622 *
623 IF( AAQQ.GT.ZERO ) THEN
624 AAPP0 = AAPP
625 *
626 * -#- M x 2 Jacobi SVD -#-
627 *
628 * -#- Safe Gram matrix computation -#-
629 *
630 IF( AAQQ.GE.ONE ) THEN
631 IF( AAPP.GE.AAQQ ) THEN
632 ROTOK = ( SMALL*AAPP ).LE.AAQQ
633 ELSE
634 ROTOK = ( SMALL*AAQQ ).LE.AAPP
635 END IF
636 IF( AAPP.LT.( BIG / AAQQ ) ) THEN
637 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
638 $ q ), 1 )*D( p )*D( q ) / AAQQ )
639 $ / AAPP
640 ELSE
641 CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
642 CALL DLASCL( 'G', 0, 0, AAPP, D( p ),
643 $ M, 1, WORK, LDA, IERR )
644 AAPQ = DDOT( M, WORK, 1, A( 1, q ),
645 $ 1 )*D( q ) / AAQQ
646 END IF
647 ELSE
648 IF( AAPP.GE.AAQQ ) THEN
649 ROTOK = AAPP.LE.( AAQQ / SMALL )
650 ELSE
651 ROTOK = AAQQ.LE.( AAPP / SMALL )
652 END IF
653 IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
654 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
655 $ q ), 1 )*D( p )*D( q ) / AAQQ )
656 $ / AAPP
657 ELSE
658 CALL DCOPY( M, A( 1, q ), 1, WORK, 1 )
659 CALL DLASCL( 'G', 0, 0, AAQQ, D( q ),
660 $ M, 1, WORK, LDA, IERR )
661 AAPQ = DDOT( M, WORK, 1, A( 1, p ),
662 $ 1 )*D( p ) / AAPP
663 END IF
664 END IF
665 *
666 MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) )
667 *
668 * TO rotate or NOT to rotate, THAT is the question ...
669 *
670 IF( DABS( AAPQ ).GT.TOL ) THEN
671 NOTROT = 0
672 * ROTATED = ROTATED + 1
673 PSKIPPED = 0
674 ISWROT = ISWROT + 1
675 *
676 IF( ROTOK ) THEN
677 *
678 AQOAP = AAQQ / AAPP
679 APOAQ = AAPP / AAQQ
680 THETA = -HALF*DABS( AQOAP-APOAQ )/AAPQ
681 IF( AAQQ.GT.AAPP0 )THETA = -THETA
682 *
683 IF( DABS( THETA ).GT.BIGTHETA ) THEN
684 T = HALF / THETA
685 FASTR( 3 ) = T*D( p ) / D( q )
686 FASTR( 4 ) = -T*D( q ) / D( p )
687 CALL DROTM( M, A( 1, p ), 1,
688 $ A( 1, q ), 1, FASTR )
689 IF( RSVEC )CALL DROTM( MVL,
690 $ V( 1, p ), 1,
691 $ V( 1, q ), 1,
692 $ FASTR )
693 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
694 $ ONE+T*APOAQ*AAPQ ) )
695 AAPP = AAPP*DSQRT( DMAX1( ZERO,
696 $ ONE-T*AQOAP*AAPQ ) )
697 MXSINJ = DMAX1( MXSINJ, DABS( T ) )
698 ELSE
699 *
700 * .. choose correct signum for THETA and rotate
701 *
702 THSIGN = -DSIGN( ONE, AAPQ )
703 IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
704 T = ONE / ( THETA+THSIGN*
705 $ DSQRT( ONE+THETA*THETA ) )
706 CS = DSQRT( ONE / ( ONE+T*T ) )
707 SN = T*CS
708 MXSINJ = DMAX1( MXSINJ, DABS( SN ) )
709 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
710 $ ONE+T*APOAQ*AAPQ ) )
711 AAPP = AAPP*DSQRT( DMAX1( ZERO,
712 $ ONE-T*AQOAP*AAPQ ) )
713 *
714 APOAQ = D( p ) / D( q )
715 AQOAP = D( q ) / D( p )
716 IF( D( p ).GE.ONE ) THEN
717 *
718 IF( D( q ).GE.ONE ) THEN
719 FASTR( 3 ) = T*APOAQ
720 FASTR( 4 ) = -T*AQOAP
721 D( p ) = D( p )*CS
722 D( q ) = D( q )*CS
723 CALL DROTM( M, A( 1, p ), 1,
724 $ A( 1, q ), 1,
725 $ FASTR )
726 IF( RSVEC )CALL DROTM( MVL,
727 $ V( 1, p ), 1, V( 1, q ),
728 $ 1, FASTR )
729 ELSE
730 CALL DAXPY( M, -T*AQOAP,
731 $ A( 1, q ), 1,
732 $ A( 1, p ), 1 )
733 CALL DAXPY( M, CS*SN*APOAQ,
734 $ A( 1, p ), 1,
735 $ A( 1, q ), 1 )
736 IF( RSVEC ) THEN
737 CALL DAXPY( MVL, -T*AQOAP,
738 $ V( 1, q ), 1,
739 $ V( 1, p ), 1 )
740 CALL DAXPY( MVL,
741 $ CS*SN*APOAQ,
742 $ V( 1, p ), 1,
743 $ V( 1, q ), 1 )
744 END IF
745 D( p ) = D( p )*CS
746 D( q ) = D( q ) / CS
747 END IF
748 ELSE
749 IF( D( q ).GE.ONE ) THEN
750 CALL DAXPY( M, T*APOAQ,
751 $ A( 1, p ), 1,
752 $ A( 1, q ), 1 )
753 CALL DAXPY( M, -CS*SN*AQOAP,
754 $ A( 1, q ), 1,
755 $ A( 1, p ), 1 )
756 IF( RSVEC ) THEN
757 CALL DAXPY( MVL, T*APOAQ,
758 $ V( 1, p ), 1,
759 $ V( 1, q ), 1 )
760 CALL DAXPY( MVL,
761 $ -CS*SN*AQOAP,
762 $ V( 1, q ), 1,
763 $ V( 1, p ), 1 )
764 END IF
765 D( p ) = D( p ) / CS
766 D( q ) = D( q )*CS
767 ELSE
768 IF( D( p ).GE.D( q ) ) THEN
769 CALL DAXPY( M, -T*AQOAP,
770 $ A( 1, q ), 1,
771 $ A( 1, p ), 1 )
772 CALL DAXPY( M, CS*SN*APOAQ,
773 $ A( 1, p ), 1,
774 $ A( 1, q ), 1 )
775 D( p ) = D( p )*CS
776 D( q ) = D( q ) / CS
777 IF( RSVEC ) THEN
778 CALL DAXPY( MVL,
779 $ -T*AQOAP,
780 $ V( 1, q ), 1,
781 $ V( 1, p ), 1 )
782 CALL DAXPY( MVL,
783 $ CS*SN*APOAQ,
784 $ V( 1, p ), 1,
785 $ V( 1, q ), 1 )
786 END IF
787 ELSE
788 CALL DAXPY( M, T*APOAQ,
789 $ A( 1, p ), 1,
790 $ A( 1, q ), 1 )
791 CALL DAXPY( M,
792 $ -CS*SN*AQOAP,
793 $ A( 1, q ), 1,
794 $ A( 1, p ), 1 )
795 D( p ) = D( p ) / CS
796 D( q ) = D( q )*CS
797 IF( RSVEC ) THEN
798 CALL DAXPY( MVL,
799 $ T*APOAQ, V( 1, p ),
800 $ 1, V( 1, q ), 1 )
801 CALL DAXPY( MVL,
802 $ -CS*SN*AQOAP,
803 $ V( 1, q ), 1,
804 $ V( 1, p ), 1 )
805 END IF
806 END IF
807 END IF
808 END IF
809 END IF
810 *
811 ELSE
812 IF( AAPP.GT.AAQQ ) THEN
813 CALL DCOPY( M, A( 1, p ), 1, WORK,
814 $ 1 )
815 CALL DLASCL( 'G', 0, 0, AAPP, ONE,
816 $ M, 1, WORK, LDA, IERR )
817 CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
818 $ M, 1, A( 1, q ), LDA,
819 $ IERR )
820 TEMP1 = -AAPQ*D( p ) / D( q )
821 CALL DAXPY( M, TEMP1, WORK, 1,
822 $ A( 1, q ), 1 )
823 CALL DLASCL( 'G', 0, 0, ONE, AAQQ,
824 $ M, 1, A( 1, q ), LDA,
825 $ IERR )
826 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
827 $ ONE-AAPQ*AAPQ ) )
828 MXSINJ = DMAX1( MXSINJ, SFMIN )
829 ELSE
830 CALL DCOPY( M, A( 1, q ), 1, WORK,
831 $ 1 )
832 CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
833 $ M, 1, WORK, LDA, IERR )
834 CALL DLASCL( 'G', 0, 0, AAPP, ONE,
835 $ M, 1, A( 1, p ), LDA,
836 $ IERR )
837 TEMP1 = -AAPQ*D( q ) / D( p )
838 CALL DAXPY( M, TEMP1, WORK, 1,
839 $ A( 1, p ), 1 )
840 CALL DLASCL( 'G', 0, 0, ONE, AAPP,
841 $ M, 1, A( 1, p ), LDA,
842 $ IERR )
843 SVA( p ) = AAPP*DSQRT( DMAX1( ZERO,
844 $ ONE-AAPQ*AAPQ ) )
845 MXSINJ = DMAX1( MXSINJ, SFMIN )
846 END IF
847 END IF
848 * END IF ROTOK THEN ... ELSE
849 *
850 * In the case of cancellation in updating SVA(q)
851 * .. recompute SVA(q)
852 IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
853 $ THEN
854 IF( ( AAQQ.LT.ROOTBIG ) .AND.
855 $ ( AAQQ.GT.ROOTSFMIN ) ) THEN
856 SVA( q ) = DNRM2( M, A( 1, q ), 1 )*
857 $ D( q )
858 ELSE
859 T = ZERO
860 AAQQ = ONE
861 CALL DLASSQ( M, A( 1, q ), 1, T,
862 $ AAQQ )
863 SVA( q ) = T*DSQRT( AAQQ )*D( q )
864 END IF
865 END IF
866 IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
867 IF( ( AAPP.LT.ROOTBIG ) .AND.
868 $ ( AAPP.GT.ROOTSFMIN ) ) THEN
869 AAPP = DNRM2( M, A( 1, p ), 1 )*
870 $ D( p )
871 ELSE
872 T = ZERO
873 AAPP = ONE
874 CALL DLASSQ( M, A( 1, p ), 1, T,
875 $ AAPP )
876 AAPP = T*DSQRT( AAPP )*D( p )
877 END IF
878 SVA( p ) = AAPP
879 END IF
880 * end of OK rotation
881 ELSE
882 NOTROT = NOTROT + 1
883 PSKIPPED = PSKIPPED + 1
884 IJBLSK = IJBLSK + 1
885 END IF
886 ELSE
887 NOTROT = NOTROT + 1
888 PSKIPPED = PSKIPPED + 1
889 IJBLSK = IJBLSK + 1
890 END IF
891 *
892 IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) )
893 $ THEN
894 SVA( p ) = AAPP
895 NOTROT = 0
896 GO TO 2011
897 END IF
898 IF( ( i.LE.SWBAND ) .AND.
899 $ ( PSKIPPED.GT.ROWSKIP ) ) THEN
900 AAPP = -AAPP
901 NOTROT = 0
902 GO TO 2203
903 END IF
904 *
905 2200 CONTINUE
906 * end of the q-loop
907 2203 CONTINUE
908 *
909 SVA( p ) = AAPP
910 *
911 ELSE
912 IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
913 $ MIN0( jgl+KBL-1, N ) - jgl + 1
914 IF( AAPP.LT.ZERO )NOTROT = 0
915 END IF
916
917 2100 CONTINUE
918 * end of the p-loop
919 2010 CONTINUE
920 * end of the jbc-loop
921 2011 CONTINUE
922 *2011 bailed out of the jbc-loop
923 DO 2012 p = igl, MIN0( igl+KBL-1, N )
924 SVA( p ) = DABS( SVA( p ) )
925 2012 CONTINUE
926 *
927 2000 CONTINUE
928 *2000 :: end of the ibr-loop
929 *
930 * .. update SVA(N)
931 IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) )
932 $ THEN
933 SVA( N ) = DNRM2( M, A( 1, N ), 1 )*D( N )
934 ELSE
935 T = ZERO
936 AAPP = ONE
937 CALL DLASSQ( M, A( 1, N ), 1, T, AAPP )
938 SVA( N ) = T*DSQRT( AAPP )*D( N )
939 END IF
940 *
941 * Additional steering devices
942 *
943 IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR.
944 $ ( ISWROT.LE.N ) ) )SWBAND = i
945 *
946 IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.DBLE( N )*TOL ) .AND.
947 $ ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN
948 GO TO 1994
949 END IF
950 *
951 IF( NOTROT.GE.EMPTSW )GO TO 1994
952
953 1993 CONTINUE
954 * end i=1:NSWEEP loop
955 * #:) Reaching this point means that the procedure has comleted the given
956 * number of iterations.
957 INFO = NSWEEP - 1
958 GO TO 1995
959 1994 CONTINUE
960 * #:) Reaching this point means that during the i-th sweep all pivots were
961 * below the given tolerance, causing early exit.
962 *
963 INFO = 0
964 * #:) INFO = 0 confirms successful iterations.
965 1995 CONTINUE
966 *
967 * Sort the vector D.
968 DO 5991 p = 1, N - 1
969 q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
970 IF( p.NE.q ) THEN
971 TEMP1 = SVA( p )
972 SVA( p ) = SVA( q )
973 SVA( q ) = TEMP1
974 TEMP1 = D( p )
975 D( p ) = D( q )
976 D( q ) = TEMP1
977 CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
978 IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
979 END IF
980 5991 CONTINUE
981 *
982 RETURN
983 * ..
984 * .. END OF DGSVJ0
985 * ..
986 END