LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
slasd6.f
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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 *
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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 occurrence 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 ( LDDIFR, 2 ) if ICOMPQ = 1 and
236 *> dimension ( K ) if ICOMPQ = 0.
237 *> On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not
238 *> defined and will not be referenced.
239 *>
240 *> If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
241 *> normalizing factors for the right singular vector matrix.
242 *>
243 *> See SLASD8 for details on DIFL and DIFR.
244 *> \endverbatim
245 *>
246 *> \param[out] Z
247 *> \verbatim
248 *> Z is REAL array, dimension ( M )
249 *> The first elements of this array contain the components
250 *> of the deflation-adjusted updating row vector.
251 *> \endverbatim
252 *>
253 *> \param[out] K
254 *> \verbatim
255 *> K is INTEGER
256 *> Contains the dimension of the non-deflated matrix,
257 *> This is the order of the related secular equation. 1 <= K <=N.
258 *> \endverbatim
259 *>
260 *> \param[out] C
261 *> \verbatim
262 *> C is REAL
263 *> C contains garbage if SQRE =0 and the C-value of a Givens
264 *> rotation related to the right null space if SQRE = 1.
265 *> \endverbatim
266 *>
267 *> \param[out] S
268 *> \verbatim
269 *> S is REAL
270 *> S contains garbage if SQRE =0 and the S-value of a Givens
271 *> rotation related to the right null space if SQRE = 1.
272 *> \endverbatim
273 *>
274 *> \param[out] WORK
275 *> \verbatim
276 *> WORK is REAL array, dimension ( 4 * M )
277 *> \endverbatim
278 *>
279 *> \param[out] IWORK
280 *> \verbatim
281 *> IWORK is INTEGER array, dimension ( 3 * N )
282 *> \endverbatim
283 *>
284 *> \param[out] INFO
285 *> \verbatim
286 *> INFO is INTEGER
287 *> = 0: successful exit.
288 *> < 0: if INFO = -i, the i-th argument had an illegal value.
289 *> > 0: if INFO = 1, a singular value did not converge
290 *> \endverbatim
291 *
292 * Authors:
293 * ========
294 *
295 *> \author Univ. of Tennessee
296 *> \author Univ. of California Berkeley
297 *> \author Univ. of Colorado Denver
298 *> \author NAG Ltd.
299 *
300 *> \date June 2016
301 *
302 *> \ingroup OTHERauxiliary
303 *
304 *> \par Contributors:
305 * ==================
306 *>
307 *> Ming Gu and Huan Ren, Computer Science Division, University of
308 *> California at Berkeley, USA
309 *>
310 * =====================================================================
311  SUBROUTINE slasd6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA,
312  $ IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM,
313  $ LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK,
314  $ IWORK, INFO )
315 *
316 * -- LAPACK auxiliary routine (version 3.7.0) --
317 * -- LAPACK is a software package provided by Univ. of Tennessee, --
318 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
319 * June 2016
320 *
321 * .. Scalar Arguments ..
322  INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
323  $ nr, sqre
324  REAL ALPHA, BETA, C, S
325 * ..
326 * .. Array Arguments ..
327  INTEGER GIVCOL( ldgcol, * ), IDXQ( * ), IWORK( * ),
328  $ perm( * )
329  REAL D( * ), DIFL( * ), DIFR( * ),
330  $ givnum( ldgnum, * ), poles( ldgnum, * ),
331  $ vf( * ), vl( * ), work( * ), z( * )
332 * ..
333 *
334 * =====================================================================
335 *
336 * .. Parameters ..
337  REAL ONE, ZERO
338  parameter( one = 1.0e+0, zero = 0.0e+0 )
339 * ..
340 * .. Local Scalars ..
341  INTEGER I, IDX, IDXC, IDXP, ISIGMA, IVFW, IVLW, IW, M,
342  $ n, n1, n2
343  REAL ORGNRM
344 * ..
345 * .. External Subroutines ..
346  EXTERNAL scopy, slamrg, slascl, slasd7, slasd8, xerbla
347 * ..
348 * .. Intrinsic Functions ..
349  INTRINSIC abs, max
350 * ..
351 * .. Executable Statements ..
352 *
353 * Test the input parameters.
354 *
355  info = 0
356  n = nl + nr + 1
357  m = n + sqre
358 *
359  IF( ( icompq.LT.0 ) .OR. ( icompq.GT.1 ) ) THEN
360  info = -1
361  ELSE IF( nl.LT.1 ) THEN
362  info = -2
363  ELSE IF( nr.LT.1 ) THEN
364  info = -3
365  ELSE IF( ( sqre.LT.0 ) .OR. ( sqre.GT.1 ) ) THEN
366  info = -4
367  ELSE IF( ldgcol.LT.n ) THEN
368  info = -14
369  ELSE IF( ldgnum.LT.n ) THEN
370  info = -16
371  END IF
372  IF( info.NE.0 ) THEN
373  CALL xerbla( 'SLASD6', -info )
374  RETURN
375  END IF
376 *
377 * The following values are for bookkeeping purposes only. They are
378 * integer pointers which indicate the portion of the workspace
379 * used by a particular array in SLASD7 and SLASD8.
380 *
381  isigma = 1
382  iw = isigma + n
383  ivfw = iw + m
384  ivlw = ivfw + m
385 *
386  idx = 1
387  idxc = idx + n
388  idxp = idxc + n
389 *
390 * Scale.
391 *
392  orgnrm = max( abs( alpha ), abs( beta ) )
393  d( nl+1 ) = zero
394  DO 10 i = 1, n
395  IF( abs( d( i ) ).GT.orgnrm ) THEN
396  orgnrm = abs( d( i ) )
397  END IF
398  10 CONTINUE
399  CALL slascl( 'G', 0, 0, orgnrm, one, n, 1, d, n, info )
400  alpha = alpha / orgnrm
401  beta = beta / orgnrm
402 *
403 * Sort and Deflate singular values.
404 *
405  CALL slasd7( icompq, nl, nr, sqre, k, d, z, work( iw ), vf,
406  $ work( ivfw ), vl, work( ivlw ), alpha, beta,
407  $ work( isigma ), iwork( idx ), iwork( idxp ), idxq,
408  $ perm, givptr, givcol, ldgcol, givnum, ldgnum, c, s,
409  $ info )
410 *
411 * Solve Secular Equation, compute DIFL, DIFR, and update VF, VL.
412 *
413  CALL slasd8( icompq, k, d, z, vf, vl, difl, difr, ldgnum,
414  $ work( isigma ), work( iw ), info )
415 *
416 * Report the possible convergence failure.
417 *
418  IF( info.NE.0 ) THEN
419  RETURN
420  END IF
421 *
422 * Save the poles if ICOMPQ = 1.
423 *
424  IF( icompq.EQ.1 ) THEN
425  CALL scopy( k, d, 1, poles( 1, 1 ), 1 )
426  CALL scopy( k, work( isigma ), 1, poles( 1, 2 ), 1 )
427  END IF
428 *
429 * Unscale.
430 *
431  CALL slascl( 'G', 0, 0, one, orgnrm, n, 1, d, n, info )
432 *
433 * Prepare the IDXQ sorting permutation.
434 *
435  n1 = k
436  n2 = n - k
437  CALL slamrg( n1, n2, d, 1, -1, idxq )
438 *
439  RETURN
440 *
441 * End of SLASD6
442 *
443  END
subroutine slasd8(ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR, DSIGMA, WORK, INFO)
SLASD8 finds the square roots of the roots of the secular equation, and stores, for each element in D...
Definition: slasd8.f:168
subroutine slasd7(ICOMPQ, NL, NR, SQRE, K, D, Z, ZW, VF, VFW, VL, VLW, ALPHA, BETA, DSIGMA, IDX, IDXP, IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S, INFO)
SLASD7 merges the two sets of singular values together into a single sorted set. Then it tries to def...
Definition: slasd7.f:282
subroutine slamrg(N1, N2, A, STRD1, STRD2, INDEX)
SLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single...
Definition: slamrg.f:101
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
SLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: slascl.f:145
subroutine slasd6(ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA, IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK, IWORK, INFO)
SLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by...
Definition: slasd6.f:315
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:84