summaryrefslogtreecommitdiff
path: root/SRC/dbdsqr.f
blob: d3630f349a54f2025fb90df86cf11ff24f7cd2be (plain)
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
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
*> \brief \b DBDSQR
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DBDSQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dbdsqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dbdsqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dbdsqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U,
*                          LDU, C, LDC, WORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   C( LDC, * ), D( * ), E( * ), U( LDU, * ),
*      $                   VT( LDVT, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DBDSQR computes the singular values and, optionally, the right and/or
*> left singular vectors from the singular value decomposition (SVD) of
*> a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
*> zero-shift QR algorithm.  The SVD of B has the form
*>
*>    B = Q * S * P**T
*>
*> where S is the diagonal matrix of singular values, Q is an orthogonal
*> matrix of left singular vectors, and P is an orthogonal matrix of
*> right singular vectors.  If left singular vectors are requested, this
*> subroutine actually returns U*Q instead of Q, and, if right singular
*> vectors are requested, this subroutine returns P**T*VT instead of
*> P**T, for given real input matrices U and VT.  When U and VT are the
*> orthogonal matrices that reduce a general matrix A to bidiagonal
*> form:  A = U*B*VT, as computed by DGEBRD, then
*>
*>    A = (U*Q) * S * (P**T*VT)
*>
*> is the SVD of A.  Optionally, the subroutine may also compute Q**T*C
*> for a given real input matrix C.
*>
*> See "Computing  Small Singular Values of Bidiagonal Matrices With
*> Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
*> LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
*> no. 5, pp. 873-912, Sept 1990) and
*> "Accurate singular values and differential qd algorithms," by
*> B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
*> Department, University of California at Berkeley, July 1992
*> for a detailed description of the algorithm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  B is upper bidiagonal;
*>          = 'L':  B is lower bidiagonal.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix B.  N >= 0.
*> \endverbatim
*>
*> \param[in] NCVT
*> \verbatim
*>          NCVT is INTEGER
*>          The number of columns of the matrix VT. NCVT >= 0.
*> \endverbatim
*>
*> \param[in] NRU
*> \verbatim
*>          NRU is INTEGER
*>          The number of rows of the matrix U. NRU >= 0.
*> \endverbatim
*>
*> \param[in] NCC
*> \verbatim
*>          NCC is INTEGER
*>          The number of columns of the matrix C. NCC >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the n diagonal elements of the bidiagonal matrix B.
*>          On exit, if INFO=0, the singular values of B in decreasing
*>          order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, the N-1 offdiagonal elements of the bidiagonal
*>          matrix B.
*>          On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
*>          will contain the diagonal and superdiagonal elements of a
*>          bidiagonal matrix orthogonally equivalent to the one given
*>          as input.
*> \endverbatim
*>
*> \param[in,out] VT
*> \verbatim
*>          VT is DOUBLE PRECISION array, dimension (LDVT, NCVT)
*>          On entry, an N-by-NCVT matrix VT.
*>          On exit, VT is overwritten by P**T * VT.
*>          Not referenced if NCVT = 0.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*>          LDVT is INTEGER
*>          The leading dimension of the array VT.
*>          LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension (LDU, N)
*>          On entry, an NRU-by-N matrix U.
*>          On exit, U is overwritten by U * Q.
*>          Not referenced if NRU = 0.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>          The leading dimension of the array U.  LDU >= max(1,NRU).
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC, NCC)
*>          On entry, an N-by-NCC matrix C.
*>          On exit, C is overwritten by Q**T * C.
*>          Not referenced if NCC = 0.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of the array C.
*>          LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  If INFO = -i, the i-th argument had an illegal value
*>          > 0:
*>             if NCVT = NRU = NCC = 0,
*>                = 1, a split was marked by a positive value in E
*>                = 2, current block of Z not diagonalized after 30*N
*>                     iterations (in inner while loop)
*>                = 3, termination criterion of outer while loop not met
*>                     (program created more than N unreduced blocks)
*>             else NCVT = NRU = NCC = 0,
*>                   the algorithm did not converge; D and E contain the
*>                   elements of a bidiagonal matrix which is orthogonally
*>                   similar to the input matrix B;  if INFO = i, i
*>                   elements of E have not converged to zero.
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  TOLMUL  DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8)))
*>          TOLMUL controls the convergence criterion of the QR loop.
*>          If it is positive, TOLMUL*EPS is the desired relative
*>             precision in the computed singular values.
*>          If it is negative, abs(TOLMUL*EPS*sigma_max) is the
*>             desired absolute accuracy in the computed singular
*>             values (corresponds to relative accuracy
*>             abs(TOLMUL*EPS) in the largest singular value.
*>          abs(TOLMUL) should be between 1 and 1/EPS, and preferably
*>             between 10 (for fast convergence) and .1/EPS
*>             (for there to be some accuracy in the results).
*>          Default is to lose at either one eighth or 2 of the
*>             available decimal digits in each computed singular value
*>             (whichever is smaller).
*>
*>  MAXITR  INTEGER, default = 6
*>          MAXITR controls the maximum number of passes of the
*>          algorithm through its inner loop. The algorithms stops
*>          (and so fails to converge) if the number of passes
*>          through the inner loop exceeds MAXITR*N**2.
*>
*> \endverbatim
*
*> \par Note:
*  ===========
*>
*> \verbatim
*>  Bug report from Cezary Dendek.
*>  On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is
*>  removed since it can overflow pretty easily (for N larger or equal
*>  than 18,919). We instead use MAXITDIVN = MAXITR*N.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup auxOTHERcomputational
*
*  =====================================================================
      SUBROUTINE DBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U,
     $                   LDU, C, LDC, WORK, INFO )
*
*  -- LAPACK computational routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   C( LDC, * ), D( * ), E( * ), U( LDU, * ),
     $                   VT( LDVT, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D0 )
      DOUBLE PRECISION   NEGONE
      PARAMETER          ( NEGONE = -1.0D0 )
      DOUBLE PRECISION   HNDRTH
      PARAMETER          ( HNDRTH = 0.01D0 )
      DOUBLE PRECISION   TEN
      PARAMETER          ( TEN = 10.0D0 )
      DOUBLE PRECISION   HNDRD
      PARAMETER          ( HNDRD = 100.0D0 )
      DOUBLE PRECISION   MEIGTH
      PARAMETER          ( MEIGTH = -0.125D0 )
      INTEGER            MAXITR
      PARAMETER          ( MAXITR = 6 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, ROTATE
      INTEGER            I, IDIR, ISUB, ITER, ITERDIVN, J, LL, LLL, M,
     $                   MAXITDIVN, NM1, NM12, NM13, OLDLL, OLDM
      DOUBLE PRECISION   ABSE, ABSS, COSL, COSR, CS, EPS, F, G, H, MU,
     $                   OLDCS, OLDSN, R, SHIFT, SIGMN, SIGMX, SINL,
     $                   SINR, SLL, SMAX, SMIN, SMINL, SMINOA,
     $                   SN, THRESH, TOL, TOLMUL, UNFL
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLARTG, DLAS2, DLASQ1, DLASR, DLASV2, DROT,
     $                   DSCAL, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, MAX, MIN, SIGN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      LOWER = LSAME( UPLO, 'L' )
      IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LOWER ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NCVT.LT.0 ) THEN
         INFO = -3
      ELSE IF( NRU.LT.0 ) THEN
         INFO = -4
      ELSE IF( NCC.LT.0 ) THEN
         INFO = -5
      ELSE IF( ( NCVT.EQ.0 .AND. LDVT.LT.1 ) .OR.
     $         ( NCVT.GT.0 .AND. LDVT.LT.MAX( 1, N ) ) ) THEN
         INFO = -9
      ELSE IF( LDU.LT.MAX( 1, NRU ) ) THEN
         INFO = -11
      ELSE IF( ( NCC.EQ.0 .AND. LDC.LT.1 ) .OR.
     $         ( NCC.GT.0 .AND. LDC.LT.MAX( 1, N ) ) ) THEN
         INFO = -13
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DBDSQR', -INFO )
         RETURN
      END IF
      IF( N.EQ.0 )
     $   RETURN
      IF( N.EQ.1 )
     $   GO TO 160
*
*     ROTATE is true if any singular vectors desired, false otherwise
*
      ROTATE = ( NCVT.GT.0 ) .OR. ( NRU.GT.0 ) .OR. ( NCC.GT.0 )
*
*     If no singular vectors desired, use qd algorithm
*
      IF( .NOT.ROTATE ) THEN
         CALL DLASQ1( N, D, E, WORK, INFO )
*
*     If INFO equals 2, dqds didn't finish, try to finish
*
         IF( INFO .NE. 2 ) RETURN
         INFO = 0
      END IF
*
      NM1 = N - 1
      NM12 = NM1 + NM1
      NM13 = NM12 + NM1
      IDIR = 0
*
*     Get machine constants
*
      EPS = DLAMCH( 'Epsilon' )
      UNFL = DLAMCH( 'Safe minimum' )
*
*     If matrix lower bidiagonal, rotate to be upper bidiagonal
*     by applying Givens rotations on the left
*
      IF( LOWER ) THEN
         DO 10 I = 1, N - 1
            CALL DLARTG( D( I ), E( I ), CS, SN, R )
            D( I ) = R
            E( I ) = SN*D( I+1 )
            D( I+1 ) = CS*D( I+1 )
            WORK( I ) = CS
            WORK( NM1+I ) = SN
   10    CONTINUE
*
*        Update singular vectors if desired
*
         IF( NRU.GT.0 )
     $      CALL DLASR( 'R', 'V', 'F', NRU, N, WORK( 1 ), WORK( N ), U,
     $                  LDU )
         IF( NCC.GT.0 )
     $      CALL DLASR( 'L', 'V', 'F', N, NCC, WORK( 1 ), WORK( N ), C,
     $                  LDC )
      END IF
*
*     Compute singular values to relative accuracy TOL
*     (By setting TOL to be negative, algorithm will compute
*     singular values to absolute accuracy ABS(TOL)*norm(input matrix))
*
      TOLMUL = MAX( TEN, MIN( HNDRD, EPS**MEIGTH ) )
      TOL = TOLMUL*EPS
*
*     Compute approximate maximum, minimum singular values
*
      SMAX = ZERO
      DO 20 I = 1, N
         SMAX = MAX( SMAX, ABS( D( I ) ) )
   20 CONTINUE
      DO 30 I = 1, N - 1
         SMAX = MAX( SMAX, ABS( E( I ) ) )
   30 CONTINUE
      SMINL = ZERO
      IF( TOL.GE.ZERO ) THEN
*
*        Relative accuracy desired
*
         SMINOA = ABS( D( 1 ) )
         IF( SMINOA.EQ.ZERO )
     $      GO TO 50
         MU = SMINOA
         DO 40 I = 2, N
            MU = ABS( D( I ) )*( MU / ( MU+ABS( E( I-1 ) ) ) )
            SMINOA = MIN( SMINOA, MU )
            IF( SMINOA.EQ.ZERO )
     $         GO TO 50
   40    CONTINUE
   50    CONTINUE
         SMINOA = SMINOA / SQRT( DBLE( N ) )
         THRESH = MAX( TOL*SMINOA, MAXITR*(N*(N*UNFL)) )
      ELSE
*
*        Absolute accuracy desired
*
         THRESH = MAX( ABS( TOL )*SMAX, MAXITR*(N*(N*UNFL)) )
      END IF
*
*     Prepare for main iteration loop for the singular values
*     (MAXIT is the maximum number of passes through the inner
*     loop permitted before nonconvergence signalled.)
*
      MAXITDIVN = MAXITR*N
      ITERDIVN = 0
      ITER = -1
      OLDLL = -1
      OLDM = -1
*
*     M points to last element of unconverged part of matrix
*
      M = N
*
*     Begin main iteration loop
*
   60 CONTINUE
*
*     Check for convergence or exceeding iteration count
*
      IF( M.LE.1 )
     $   GO TO 160
*
      IF( ITER.GE.N ) THEN
         ITER = ITER - N
         ITERDIVN = ITERDIVN + 1
         IF( ITERDIVN.GE.MAXITDIVN )
     $      GO TO 200
      END IF
*
*     Find diagonal block of matrix to work on
*
      IF( TOL.LT.ZERO .AND. ABS( D( M ) ).LE.THRESH )
     $   D( M ) = ZERO
      SMAX = ABS( D( M ) )
      SMIN = SMAX
      DO 70 LLL = 1, M - 1
         LL = M - LLL
         ABSS = ABS( D( LL ) )
         ABSE = ABS( E( LL ) )
         IF( TOL.LT.ZERO .AND. ABSS.LE.THRESH )
     $      D( LL ) = ZERO
         IF( ABSE.LE.THRESH )
     $      GO TO 80
         SMIN = MIN( SMIN, ABSS )
         SMAX = MAX( SMAX, ABSS, ABSE )
   70 CONTINUE
      LL = 0
      GO TO 90
   80 CONTINUE
      E( LL ) = ZERO
*
*     Matrix splits since E(LL) = 0
*
      IF( LL.EQ.M-1 ) THEN
*
*        Convergence of bottom singular value, return to top of loop
*
         M = M - 1
         GO TO 60
      END IF
   90 CONTINUE
      LL = LL + 1
*
*     E(LL) through E(M-1) are nonzero, E(LL-1) is zero
*
      IF( LL.EQ.M-1 ) THEN
*
*        2 by 2 block, handle separately
*
         CALL DLASV2( D( M-1 ), E( M-1 ), D( M ), SIGMN, SIGMX, SINR,
     $                COSR, SINL, COSL )
         D( M-1 ) = SIGMX
         E( M-1 ) = ZERO
         D( M ) = SIGMN
*
*        Compute singular vectors, if desired
*
         IF( NCVT.GT.0 )
     $      CALL DROT( NCVT, VT( M-1, 1 ), LDVT, VT( M, 1 ), LDVT, COSR,
     $                 SINR )
         IF( NRU.GT.0 )
     $      CALL DROT( NRU, U( 1, M-1 ), 1, U( 1, M ), 1, COSL, SINL )
         IF( NCC.GT.0 )
     $      CALL DROT( NCC, C( M-1, 1 ), LDC, C( M, 1 ), LDC, COSL,
     $                 SINL )
         M = M - 2
         GO TO 60
      END IF
*
*     If working on new submatrix, choose shift direction
*     (from larger end diagonal element towards smaller)
*
      IF( LL.GT.OLDM .OR. M.LT.OLDLL ) THEN
         IF( ABS( D( LL ) ).GE.ABS( D( M ) ) ) THEN
*
*           Chase bulge from top (big end) to bottom (small end)
*
            IDIR = 1
         ELSE
*
*           Chase bulge from bottom (big end) to top (small end)
*
            IDIR = 2
         END IF
      END IF
*
*     Apply convergence tests
*
      IF( IDIR.EQ.1 ) THEN
*
*        Run convergence test in forward direction
*        First apply standard test to bottom of matrix
*
         IF( ABS( E( M-1 ) ).LE.ABS( TOL )*ABS( D( M ) ) .OR.
     $       ( TOL.LT.ZERO .AND. ABS( E( M-1 ) ).LE.THRESH ) ) THEN
            E( M-1 ) = ZERO
            GO TO 60
         END IF
*
         IF( TOL.GE.ZERO ) THEN
*
*           If relative accuracy desired,
*           apply convergence criterion forward
*
            MU = ABS( D( LL ) )
            SMINL = MU
            DO 100 LLL = LL, M - 1
               IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN
                  E( LLL ) = ZERO
                  GO TO 60
               END IF
               MU = ABS( D( LLL+1 ) )*( MU / ( MU+ABS( E( LLL ) ) ) )
               SMINL = MIN( SMINL, MU )
  100       CONTINUE
         END IF
*
      ELSE
*
*        Run convergence test in backward direction
*        First apply standard test to top of matrix
*
         IF( ABS( E( LL ) ).LE.ABS( TOL )*ABS( D( LL ) ) .OR.
     $       ( TOL.LT.ZERO .AND. ABS( E( LL ) ).LE.THRESH ) ) THEN
            E( LL ) = ZERO
            GO TO 60
         END IF
*
         IF( TOL.GE.ZERO ) THEN
*
*           If relative accuracy desired,
*           apply convergence criterion backward
*
            MU = ABS( D( M ) )
            SMINL = MU
            DO 110 LLL = M - 1, LL, -1
               IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN
                  E( LLL ) = ZERO
                  GO TO 60
               END IF
               MU = ABS( D( LLL ) )*( MU / ( MU+ABS( E( LLL ) ) ) )
               SMINL = MIN( SMINL, MU )
  110       CONTINUE
         END IF
      END IF
      OLDLL = LL
      OLDM = M
*
*     Compute shift.  First, test if shifting would ruin relative
*     accuracy, and if so set the shift to zero.
*
      IF( TOL.GE.ZERO .AND. N*TOL*( SMINL / SMAX ).LE.
     $    MAX( EPS, HNDRTH*TOL ) ) THEN
*
*        Use a zero shift to avoid loss of relative accuracy
*
         SHIFT = ZERO
      ELSE
*
*        Compute the shift from 2-by-2 block at end of matrix
*
         IF( IDIR.EQ.1 ) THEN
            SLL = ABS( D( LL ) )
            CALL DLAS2( D( M-1 ), E( M-1 ), D( M ), SHIFT, R )
         ELSE
            SLL = ABS( D( M ) )
            CALL DLAS2( D( LL ), E( LL ), D( LL+1 ), SHIFT, R )
         END IF
*
*        Test if shift negligible, and if so set to zero
*
         IF( SLL.GT.ZERO ) THEN
            IF( ( SHIFT / SLL )**2.LT.EPS )
     $         SHIFT = ZERO
         END IF
      END IF
*
*     Increment iteration count
*
      ITER = ITER + M - LL
*
*     If SHIFT = 0, do simplified QR iteration
*
      IF( SHIFT.EQ.ZERO ) THEN
         IF( IDIR.EQ.1 ) THEN
*
*           Chase bulge from top to bottom
*           Save cosines and sines for later singular vector updates
*
            CS = ONE
            OLDCS = ONE
            DO 120 I = LL, M - 1
               CALL DLARTG( D( I )*CS, E( I ), CS, SN, R )
               IF( I.GT.LL )
     $            E( I-1 ) = OLDSN*R
               CALL DLARTG( OLDCS*R, D( I+1 )*SN, OLDCS, OLDSN, D( I ) )
               WORK( I-LL+1 ) = CS
               WORK( I-LL+1+NM1 ) = SN
               WORK( I-LL+1+NM12 ) = OLDCS
               WORK( I-LL+1+NM13 ) = OLDSN
  120       CONTINUE
            H = D( M )*CS
            D( M ) = H*OLDCS
            E( M-1 ) = H*OLDSN
*
*           Update singular vectors
*
            IF( NCVT.GT.0 )
     $         CALL DLASR( 'L', 'V', 'F', M-LL+1, NCVT, WORK( 1 ),
     $                     WORK( N ), VT( LL, 1 ), LDVT )
            IF( NRU.GT.0 )
     $         CALL DLASR( 'R', 'V', 'F', NRU, M-LL+1, WORK( NM12+1 ),
     $                     WORK( NM13+1 ), U( 1, LL ), LDU )
            IF( NCC.GT.0 )
     $         CALL DLASR( 'L', 'V', 'F', M-LL+1, NCC, WORK( NM12+1 ),
     $                     WORK( NM13+1 ), C( LL, 1 ), LDC )
*
*           Test convergence
*
            IF( ABS( E( M-1 ) ).LE.THRESH )
     $         E( M-1 ) = ZERO
*
         ELSE
*
*           Chase bulge from bottom to top
*           Save cosines and sines for later singular vector updates
*
            CS = ONE
            OLDCS = ONE
            DO 130 I = M, LL + 1, -1
               CALL DLARTG( D( I )*CS, E( I-1 ), CS, SN, R )
               IF( I.LT.M )
     $            E( I ) = OLDSN*R
               CALL DLARTG( OLDCS*R, D( I-1 )*SN, OLDCS, OLDSN, D( I ) )
               WORK( I-LL ) = CS
               WORK( I-LL+NM1 ) = -SN
               WORK( I-LL+NM12 ) = OLDCS
               WORK( I-LL+NM13 ) = -OLDSN
  130       CONTINUE
            H = D( LL )*CS
            D( LL ) = H*OLDCS
            E( LL ) = H*OLDSN
*
*           Update singular vectors
*
            IF( NCVT.GT.0 )
     $         CALL DLASR( 'L', 'V', 'B', M-LL+1, NCVT, WORK( NM12+1 ),
     $                     WORK( NM13+1 ), VT( LL, 1 ), LDVT )
            IF( NRU.GT.0 )
     $         CALL DLASR( 'R', 'V', 'B', NRU, M-LL+1, WORK( 1 ),
     $                     WORK( N ), U( 1, LL ), LDU )
            IF( NCC.GT.0 )
     $         CALL DLASR( 'L', 'V', 'B', M-LL+1, NCC, WORK( 1 ),
     $                     WORK( N ), C( LL, 1 ), LDC )
*
*           Test convergence
*
            IF( ABS( E( LL ) ).LE.THRESH )
     $         E( LL ) = ZERO
         END IF
      ELSE
*
*        Use nonzero shift
*
         IF( IDIR.EQ.1 ) THEN
*
*           Chase bulge from top to bottom
*           Save cosines and sines for later singular vector updates
*
            F = ( ABS( D( LL ) )-SHIFT )*
     $          ( SIGN( ONE, D( LL ) )+SHIFT / D( LL ) )
            G = E( LL )
            DO 140 I = LL, M - 1
               CALL DLARTG( F, G, COSR, SINR, R )
               IF( I.GT.LL )
     $            E( I-1 ) = R
               F = COSR*D( I ) + SINR*E( I )
               E( I ) = COSR*E( I ) - SINR*D( I )
               G = SINR*D( I+1 )
               D( I+1 ) = COSR*D( I+1 )
               CALL DLARTG( F, G, COSL, SINL, R )
               D( I ) = R
               F = COSL*E( I ) + SINL*D( I+1 )
               D( I+1 ) = COSL*D( I+1 ) - SINL*E( I )
               IF( I.LT.M-1 ) THEN
                  G = SINL*E( I+1 )
                  E( I+1 ) = COSL*E( I+1 )
               END IF
               WORK( I-LL+1 ) = COSR
               WORK( I-LL+1+NM1 ) = SINR
               WORK( I-LL+1+NM12 ) = COSL
               WORK( I-LL+1+NM13 ) = SINL
  140       CONTINUE
            E( M-1 ) = F
*
*           Update singular vectors
*
            IF( NCVT.GT.0 )
     $         CALL DLASR( 'L', 'V', 'F', M-LL+1, NCVT, WORK( 1 ),
     $                     WORK( N ), VT( LL, 1 ), LDVT )
            IF( NRU.GT.0 )
     $         CALL DLASR( 'R', 'V', 'F', NRU, M-LL+1, WORK( NM12+1 ),
     $                     WORK( NM13+1 ), U( 1, LL ), LDU )
            IF( NCC.GT.0 )
     $         CALL DLASR( 'L', 'V', 'F', M-LL+1, NCC, WORK( NM12+1 ),
     $                     WORK( NM13+1 ), C( LL, 1 ), LDC )
*
*           Test convergence
*
            IF( ABS( E( M-1 ) ).LE.THRESH )
     $         E( M-1 ) = ZERO
*
         ELSE
*
*           Chase bulge from bottom to top
*           Save cosines and sines for later singular vector updates
*
            F = ( ABS( D( M ) )-SHIFT )*( SIGN( ONE, D( M ) )+SHIFT /
     $          D( M ) )
            G = E( M-1 )
            DO 150 I = M, LL + 1, -1
               CALL DLARTG( F, G, COSR, SINR, R )
               IF( I.LT.M )
     $            E( I ) = R
               F = COSR*D( I ) + SINR*E( I-1 )
               E( I-1 ) = COSR*E( I-1 ) - SINR*D( I )
               G = SINR*D( I-1 )
               D( I-1 ) = COSR*D( I-1 )
               CALL DLARTG( F, G, COSL, SINL, R )
               D( I ) = R
               F = COSL*E( I-1 ) + SINL*D( I-1 )
               D( I-1 ) = COSL*D( I-1 ) - SINL*E( I-1 )
               IF( I.GT.LL+1 ) THEN
                  G = SINL*E( I-2 )
                  E( I-2 ) = COSL*E( I-2 )
               END IF
               WORK( I-LL ) = COSR
               WORK( I-LL+NM1 ) = -SINR
               WORK( I-LL+NM12 ) = COSL
               WORK( I-LL+NM13 ) = -SINL
  150       CONTINUE
            E( LL ) = F
*
*           Test convergence
*
            IF( ABS( E( LL ) ).LE.THRESH )
     $         E( LL ) = ZERO
*
*           Update singular vectors if desired
*
            IF( NCVT.GT.0 )
     $         CALL DLASR( 'L', 'V', 'B', M-LL+1, NCVT, WORK( NM12+1 ),
     $                     WORK( NM13+1 ), VT( LL, 1 ), LDVT )
            IF( NRU.GT.0 )
     $         CALL DLASR( 'R', 'V', 'B', NRU, M-LL+1, WORK( 1 ),
     $                     WORK( N ), U( 1, LL ), LDU )
            IF( NCC.GT.0 )
     $         CALL DLASR( 'L', 'V', 'B', M-LL+1, NCC, WORK( 1 ),
     $                     WORK( N ), C( LL, 1 ), LDC )
         END IF
      END IF
*
*     QR iteration finished, go back and check convergence
*
      GO TO 60
*
*     All singular values converged, so make them positive
*
  160 CONTINUE
      DO 170 I = 1, N
         IF( D( I ).LT.ZERO ) THEN
            D( I ) = -D( I )
*
*           Change sign of singular vectors, if desired
*
            IF( NCVT.GT.0 )
     $         CALL DSCAL( NCVT, NEGONE, VT( I, 1 ), LDVT )
         END IF
  170 CONTINUE
*
*     Sort the singular values into decreasing order (insertion sort on
*     singular values, but only one transposition per singular vector)
*
      DO 190 I = 1, N - 1
*
*        Scan for smallest D(I)
*
         ISUB = 1
         SMIN = D( 1 )
         DO 180 J = 2, N + 1 - I
            IF( D( J ).LE.SMIN ) THEN
               ISUB = J
               SMIN = D( J )
            END IF
  180    CONTINUE
         IF( ISUB.NE.N+1-I ) THEN
*
*           Swap singular values and vectors
*
            D( ISUB ) = D( N+1-I )
            D( N+1-I ) = SMIN
            IF( NCVT.GT.0 )
     $         CALL DSWAP( NCVT, VT( ISUB, 1 ), LDVT, VT( N+1-I, 1 ),
     $                     LDVT )
            IF( NRU.GT.0 )
     $         CALL DSWAP( NRU, U( 1, ISUB ), 1, U( 1, N+1-I ), 1 )
            IF( NCC.GT.0 )
     $         CALL DSWAP( NCC, C( ISUB, 1 ), LDC, C( N+1-I, 1 ), LDC )
         END IF
  190 CONTINUE
      GO TO 220
*
*     Maximum number of iterations exceeded, failure to converge
*
  200 CONTINUE
      INFO = 0
      DO 210 I = 1, N - 1
         IF( E( I ).NE.ZERO )
     $      INFO = INFO + 1
  210 CONTINUE
  220 CONTINUE
      RETURN
*
*     End of DBDSQR
*
      END