1*> \brief \b SLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc.
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slasd6.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE SLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA,
22*                          IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM,
23*                          LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK,
24*                          IWORK, INFO )
25*
26*       .. Scalar Arguments ..
27*       INTEGER            GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
28*      $                   NR, SQRE
29*       REAL               ALPHA, BETA, C, S
30*       ..
31*       .. Array Arguments ..
32*       INTEGER            GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ),
33*      $                   PERM( * )
34*       REAL               D( * ), DIFL( * ), DIFR( * ),
35*      $                   GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
36*      $                   VF( * ), VL( * ), WORK( * ), Z( * )
37*       ..
38*
39*
40*> \par Purpose:
41*  =============
42*>
43*> \verbatim
44*>
45*> SLASD6 computes the SVD of an updated upper bidiagonal matrix B
46*> obtained by merging two smaller ones by appending a row. This
47*> routine is used only for the problem which requires all singular
48*> values and optionally singular vector matrices in factored form.
49*> B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE.
50*> A related subroutine, SLASD1, handles the case in which all singular
51*> values and singular vectors of the bidiagonal matrix are desired.
52*>
53*> SLASD6 computes the SVD as follows:
54*>
55*>               ( D1(in)    0    0       0 )
56*>   B = U(in) * (   Z1**T   a   Z2**T    b ) * VT(in)
57*>               (   0       0   D2(in)   0 )
58*>
59*>     = U(out) * ( D(out) 0) * VT(out)
60*>
61*> where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M
62*> with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
63*> elsewhere; and the entry b is empty if SQRE = 0.
64*>
65*> The singular values of B can be computed using D1, D2, the first
66*> components of all the right singular vectors of the lower block, and
67*> the last components of all the right singular vectors of the upper
68*> block. These components are stored and updated in VF and VL,
69*> respectively, in SLASD6. Hence U and VT are not explicitly
70*> referenced.
71*>
72*> The singular values are stored in D. The algorithm consists of two
73*> stages:
74*>
75*>       The first stage consists of deflating the size of the problem
76*>       when there are multiple singular values or if there is a zero
77*>       in the Z vector. For each such occurence the dimension of the
78*>       secular equation problem is reduced by one. This stage is
79*>       performed by the routine SLASD7.
80*>
81*>       The second stage consists of calculating the updated
82*>       singular values. This is done by finding the roots of the
83*>       secular equation via the routine SLASD4 (as called by SLASD8).
84*>       This routine also updates VF and VL and computes the distances
85*>       between the updated singular values and the old singular
86*>       values.
87*>
88*> SLASD6 is called from SLASDA.
89*> \endverbatim
90*
91*  Arguments:
92*  ==========
93*
94*> \param[in] ICOMPQ
95*> \verbatim
96*>          ICOMPQ is INTEGER
97*>         Specifies whether singular vectors are to be computed in
98*>         factored form:
99*>         = 0: Compute singular values only.
100*>         = 1: Compute singular vectors in factored form as well.
101*> \endverbatim
102*>
103*> \param[in] NL
104*> \verbatim
105*>          NL is INTEGER
106*>         The row dimension of the upper block.  NL >= 1.
107*> \endverbatim
108*>
109*> \param[in] NR
110*> \verbatim
111*>          NR is INTEGER
112*>         The row dimension of the lower block.  NR >= 1.
113*> \endverbatim
114*>
115*> \param[in] SQRE
116*> \verbatim
117*>          SQRE is INTEGER
118*>         = 0: the lower block is an NR-by-NR square matrix.
119*>         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
120*>
121*>         The bidiagonal matrix has row dimension N = NL + NR + 1,
122*>         and column dimension M = N + SQRE.
123*> \endverbatim
124*>
125*> \param[in,out] D
126*> \verbatim
127*>          D is REAL array, dimension (NL+NR+1).
128*>         On entry D(1:NL,1:NL) contains the singular values of the
129*>         upper block, and D(NL+2:N) contains the singular values
130*>         of the lower block. On exit D(1:N) contains the singular
131*>         values of the modified matrix.
132*> \endverbatim
133*>
134*> \param[in,out] VF
135*> \verbatim
136*>          VF is REAL array, dimension (M)
137*>         On entry, VF(1:NL+1) contains the first components of all
138*>         right singular vectors of the upper block; and VF(NL+2:M)
139*>         contains the first components of all right singular vectors
140*>         of the lower block. On exit, VF contains the first components
141*>         of all right singular vectors of the bidiagonal matrix.
142*> \endverbatim
143*>
144*> \param[in,out] VL
145*> \verbatim
146*>          VL is REAL array, dimension (M)
147*>         On entry, VL(1:NL+1) contains the  last components of all
148*>         right singular vectors of the upper block; and VL(NL+2:M)
149*>         contains the last components of all right singular vectors of
150*>         the lower block. On exit, VL contains the last components of
151*>         all right singular vectors of the bidiagonal matrix.
152*> \endverbatim
153*>
154*> \param[in,out] ALPHA
155*> \verbatim
156*>          ALPHA is REAL
157*>         Contains the diagonal element associated with the added row.
158*> \endverbatim
159*>
160*> \param[in,out] BETA
161*> \verbatim
162*>          BETA is REAL
163*>         Contains the off-diagonal element associated with the added
164*>         row.
165*> \endverbatim
166*>
167*> \param[in,out] IDXQ
168*> \verbatim
169*>          IDXQ is INTEGER array, dimension (N)
170*>         This contains the permutation which will reintegrate the
171*>         subproblem just solved back into sorted order, i.e.
172*>         D( IDXQ( I = 1, N ) ) will be in ascending order.
173*> \endverbatim
174*>
175*> \param[out] PERM
176*> \verbatim
177*>          PERM is INTEGER array, dimension ( N )
178*>         The permutations (from deflation and sorting) to be applied
179*>         to each block. Not referenced if ICOMPQ = 0.
180*> \endverbatim
181*>
182*> \param[out] GIVPTR
183*> \verbatim
184*>          GIVPTR is INTEGER
185*>         The number of Givens rotations which took place in this
186*>         subproblem. Not referenced if ICOMPQ = 0.
187*> \endverbatim
188*>
189*> \param[out] GIVCOL
190*> \verbatim
191*>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
192*>         Each pair of numbers indicates a pair of columns to take place
193*>         in a Givens rotation. Not referenced if ICOMPQ = 0.
194*> \endverbatim
195*>
196*> \param[in] LDGCOL
197*> \verbatim
198*>          LDGCOL is INTEGER
199*>         leading dimension of GIVCOL, must be at least N.
200*> \endverbatim
201*>
202*> \param[out] GIVNUM
203*> \verbatim
204*>          GIVNUM is REAL array, dimension ( LDGNUM, 2 )
205*>         Each number indicates the C or S value to be used in the
206*>         corresponding Givens rotation. Not referenced if ICOMPQ = 0.
207*> \endverbatim
208*>
209*> \param[in] LDGNUM
210*> \verbatim
211*>          LDGNUM is INTEGER
212*>         The leading dimension of GIVNUM and POLES, must be at least N.
213*> \endverbatim
214*>
215*> \param[out] POLES
216*> \verbatim
217*>          POLES is REAL array, dimension ( LDGNUM, 2 )
218*>         On exit, POLES(1,*) is an array containing the new singular
219*>         values obtained from solving the secular equation, and
220*>         POLES(2,*) is an array containing the poles in the secular
221*>         equation. Not referenced if ICOMPQ = 0.
222*> \endverbatim
223*>
224*> \param[out] DIFL
225*> \verbatim
226*>          DIFL is REAL array, dimension ( N )
227*>         On exit, DIFL(I) is the distance between I-th updated
228*>         (undeflated) singular value and the I-th (undeflated) old
229*>         singular value.
230*> \endverbatim
231*>
232*> \param[out] DIFR
233*> \verbatim
234*>          DIFR is REAL array,
235*>                  dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and
236*>                  dimension ( N ) if ICOMPQ = 0.
237*>         On exit, DIFR(I, 1) is the distance between I-th updated
238*>         (undeflated) singular value and the I+1-th (undeflated) old
239*>         singular value.
240*>
241*>         If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
242*>         normalizing factors for the right singular vector matrix.
243*>
244*>         See SLASD8 for details on DIFL and DIFR.
245*> \endverbatim
246*>
247*> \param[out] Z
248*> \verbatim
249*>          Z is REAL array, dimension ( M )
250*>         The first elements of this array contain the components
251*>         of the deflation-adjusted updating row vector.
252*> \endverbatim
253*>
254*> \param[out] K
255*> \verbatim
256*>          K is INTEGER
257*>         Contains the dimension of the non-deflated matrix,
258*>         This is the order of the related secular equation. 1 <= K <=N.
259*> \endverbatim
260*>
261*> \param[out] C
262*> \verbatim
263*>          C is REAL
264*>         C contains garbage if SQRE =0 and the C-value of a Givens
265*>         rotation related to the right null space if SQRE = 1.
266*> \endverbatim
267*>
268*> \param[out] S
269*> \verbatim
270*>          S is REAL
271*>         S contains garbage if SQRE =0 and the S-value of a Givens
272*>         rotation related to the right null space if SQRE = 1.
273*> \endverbatim
274*>
275*> \param[out] WORK
276*> \verbatim
277*>          WORK is REAL array, dimension ( 4 * M )
278*> \endverbatim
279*>
280*> \param[out] IWORK
281*> \verbatim
282*>          IWORK is INTEGER array, dimension ( 3 * N )
283*> \endverbatim
284*>
285*> \param[out] INFO
286*> \verbatim
287*>          INFO is INTEGER
288*>          = 0:  successful exit.
289*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
290*>          > 0:  if INFO = 1, a singular value did not converge
291*> \endverbatim
292*
293*  Authors:
294*  ========
295*
296*> \author Univ. of Tennessee
297*> \author Univ. of California Berkeley
298*> \author Univ. of Colorado Denver
299*> \author NAG Ltd.
300*
301*> \date November 2015
302*
303*> \ingroup auxOTHERauxiliary
304*
305*> \par Contributors:
306*  ==================
307*>
308*>     Ming Gu and Huan Ren, Computer Science Division, University of
309*>     California at Berkeley, USA
310*>
311*  =====================================================================
312      SUBROUTINE SLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA,
313     $                   IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM,
314     $                   LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK,
315     $                   IWORK, INFO )
316*
317*  -- LAPACK auxiliary routine (version 3.6.0) --
318*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
319*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
320*     November 2015
321*
322*     .. Scalar Arguments ..
323      INTEGER            GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
324     $                   NR, SQRE
325      REAL               ALPHA, BETA, C, S
326*     ..
327*     .. Array Arguments ..
328      INTEGER            GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ),
329     $                   PERM( * )
330      REAL               D( * ), DIFL( * ), DIFR( * ),
331     $                   GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
332     $                   VF( * ), VL( * ), WORK( * ), Z( * )
333*     ..
334*
335*  =====================================================================
336*
337*     .. Parameters ..
338      REAL               ONE, ZERO
339      PARAMETER          ( ONE = 1.0E+0, ZERO = 0.0E+0 )
340*     ..
341*     .. Local Scalars ..
342      INTEGER            I, IDX, IDXC, IDXP, ISIGMA, IVFW, IVLW, IW, M,
343     $                   N, N1, N2
344      REAL               ORGNRM
345*     ..
346*     .. External Subroutines ..
347      EXTERNAL           SCOPY, SLAMRG, SLASCL, SLASD7, SLASD8, XERBLA
348*     ..
349*     .. Intrinsic Functions ..
350      INTRINSIC          ABS, MAX
351*     ..
352*     .. Executable Statements ..
353*
354*     Test the input parameters.
355*
356      INFO = 0
357      N = NL + NR + 1
358      M = N + SQRE
359*
360      IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
361         INFO = -1
362      ELSE IF( NL.LT.1 ) THEN
363         INFO = -2
364      ELSE IF( NR.LT.1 ) THEN
365         INFO = -3
366      ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
367         INFO = -4
368      ELSE IF( LDGCOL.LT.N ) THEN
369         INFO = -14
370      ELSE IF( LDGNUM.LT.N ) THEN
371         INFO = -16
372      END IF
373      IF( INFO.NE.0 ) THEN
374         CALL XERBLA( 'SLASD6', -INFO )
375         RETURN
376      END IF
377*
378*     The following values are for bookkeeping purposes only.  They are
379*     integer pointers which indicate the portion of the workspace
380*     used by a particular array in SLASD7 and SLASD8.
381*
382      ISIGMA = 1
383      IW = ISIGMA + N
384      IVFW = IW + M
385      IVLW = IVFW + M
386*
387      IDX = 1
388      IDXC = IDX + N
389      IDXP = IDXC + N
390*
391*     Scale.
392*
393      ORGNRM = MAX( ABS( ALPHA ), ABS( BETA ) )
394      D( NL+1 ) = ZERO
395      DO 10 I = 1, N
396         IF( ABS( D( I ) ).GT.ORGNRM ) THEN
397            ORGNRM = ABS( D( I ) )
398         END IF
399   10 CONTINUE
400      CALL SLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
401      ALPHA = ALPHA / ORGNRM
402      BETA = BETA / ORGNRM
403*
404*     Sort and Deflate singular values.
405*
406      CALL SLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, WORK( IW ), VF,
407     $             WORK( IVFW ), VL, WORK( IVLW ), ALPHA, BETA,
408     $             WORK( ISIGMA ), IWORK( IDX ), IWORK( IDXP ), IDXQ,
409     $             PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S,
410     $             INFO )
411*
412*     Solve Secular Equation, compute DIFL, DIFR, and update VF, VL.
413*
414      CALL SLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDGNUM,
415     $             WORK( ISIGMA ), WORK( IW ), INFO )
416*
417*     Report the possible convergence failure.
418*
419      IF( INFO.NE.0 ) THEN
420         RETURN
421      END IF
422*
423*     Save the poles if ICOMPQ = 1.
424*
425      IF( ICOMPQ.EQ.1 ) THEN
426         CALL SCOPY( K, D, 1, POLES( 1, 1 ), 1 )
427         CALL SCOPY( K, WORK( ISIGMA ), 1, POLES( 1, 2 ), 1 )
428      END IF
429*
430*     Unscale.
431*
432      CALL SLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
433*
434*     Prepare the IDXQ sorting permutation.
435*
436      N1 = K
437      N2 = N - K
438      CALL SLAMRG( N1, N2, D, 1, -1, IDXQ )
439*
440      RETURN
441*
442*     End of SLASD6
443*
444      END
445