LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
slasd1.f
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1 *> \brief \b SLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.
2 *
3 * =========== DOCUMENTATION ===========
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT,
22 * IDXQ, IWORK, WORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * INTEGER INFO, LDU, LDVT, NL, NR, SQRE
26 * REAL ALPHA, BETA
27 * ..
28 * .. Array Arguments ..
29 * INTEGER IDXQ( * ), IWORK( * )
30 * REAL D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> SLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B,
40 *> where N = NL + NR + 1 and M = N + SQRE. SLASD1 is called from SLASD0.
41 *>
42 *> A related subroutine SLASD7 handles the case in which the singular
43 *> values (and the singular vectors in factored form) are desired.
44 *>
45 *> SLASD1 computes the SVD as follows:
46 *>
47 *> ( D1(in) 0 0 0 )
48 *> B = U(in) * ( Z1**T a Z2**T b ) * VT(in)
49 *> ( 0 0 D2(in) 0 )
50 *>
51 *> = U(out) * ( D(out) 0) * VT(out)
52 *>
53 *> where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M
54 *> with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
55 *> elsewhere; and the entry b is empty if SQRE = 0.
56 *>
57 *> The left singular vectors of the original matrix are stored in U, and
58 *> the transpose of the right singular vectors are stored in VT, and the
59 *> singular values are in D. The algorithm consists of three stages:
60 *>
61 *> The first stage consists of deflating the size of the problem
62 *> when there are multiple singular values or when there are zeros in
63 *> the Z vector. For each such occurrence the dimension of the
64 *> secular equation problem is reduced by one. This stage is
65 *> performed by the routine SLASD2.
66 *>
67 *> The second stage consists of calculating the updated
68 *> singular values. This is done by finding the square roots of the
69 *> roots of the secular equation via the routine SLASD4 (as called
70 *> by SLASD3). This routine also calculates the singular vectors of
71 *> the current problem.
72 *>
73 *> The final stage consists of computing the updated singular vectors
74 *> directly using the updated singular values. The singular vectors
75 *> for the current problem are multiplied with the singular vectors
76 *> from the overall problem.
77 *> \endverbatim
78 *
79 * Arguments:
80 * ==========
81 *
82 *> \param[in] NL
83 *> \verbatim
84 *> NL is INTEGER
85 *> The row dimension of the upper block. NL >= 1.
86 *> \endverbatim
87 *>
88 *> \param[in] NR
89 *> \verbatim
90 *> NR is INTEGER
91 *> The row dimension of the lower block. NR >= 1.
92 *> \endverbatim
93 *>
94 *> \param[in] SQRE
95 *> \verbatim
96 *> SQRE is INTEGER
97 *> = 0: the lower block is an NR-by-NR square matrix.
98 *> = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
99 *>
100 *> The bidiagonal matrix has row dimension N = NL + NR + 1,
101 *> and column dimension M = N + SQRE.
102 *> \endverbatim
103 *>
104 *> \param[in,out] D
105 *> \verbatim
106 *> D is REAL array, dimension (NL+NR+1).
107 *> N = NL+NR+1
108 *> On entry D(1:NL,1:NL) contains the singular values of the
109 *> upper block; and D(NL+2:N) contains the singular values of
110 *> the lower block. On exit D(1:N) contains the singular values
111 *> of the modified matrix.
112 *> \endverbatim
113 *>
114 *> \param[in,out] ALPHA
115 *> \verbatim
116 *> ALPHA is REAL
117 *> Contains the diagonal element associated with the added row.
118 *> \endverbatim
119 *>
120 *> \param[in,out] BETA
121 *> \verbatim
122 *> BETA is REAL
123 *> Contains the off-diagonal element associated with the added
124 *> row.
125 *> \endverbatim
126 *>
127 *> \param[in,out] U
128 *> \verbatim
129 *> U is REAL array, dimension (LDU,N)
130 *> On entry U(1:NL, 1:NL) contains the left singular vectors of
131 *> the upper block; U(NL+2:N, NL+2:N) contains the left singular
132 *> vectors of the lower block. On exit U contains the left
133 *> singular vectors of the bidiagonal matrix.
134 *> \endverbatim
135 *>
136 *> \param[in] LDU
137 *> \verbatim
138 *> LDU is INTEGER
139 *> The leading dimension of the array U. LDU >= max( 1, N ).
140 *> \endverbatim
141 *>
142 *> \param[in,out] VT
143 *> \verbatim
144 *> VT is REAL array, dimension (LDVT,M)
145 *> where M = N + SQRE.
146 *> On entry VT(1:NL+1, 1:NL+1)**T contains the right singular
147 *> vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains
148 *> the right singular vectors of the lower block. On exit
149 *> VT**T contains the right singular vectors of the
150 *> bidiagonal matrix.
151 *> \endverbatim
152 *>
153 *> \param[in] LDVT
154 *> \verbatim
155 *> LDVT is INTEGER
156 *> The leading dimension of the array VT. LDVT >= max( 1, M ).
157 *> \endverbatim
158 *>
159 *> \param[in,out] IDXQ
160 *> \verbatim
161 *> IDXQ is INTEGER array, dimension (N)
162 *> This contains the permutation which will reintegrate the
163 *> subproblem just solved back into sorted order, i.e.
164 *> D( IDXQ( I = 1, N ) ) will be in ascending order.
165 *> \endverbatim
166 *>
167 *> \param[out] IWORK
168 *> \verbatim
169 *> IWORK is INTEGER array, dimension (4*N)
170 *> \endverbatim
171 *>
172 *> \param[out] WORK
173 *> \verbatim
174 *> WORK is REAL array, dimension (3*M**2+2*M)
175 *> \endverbatim
176 *>
177 *> \param[out] INFO
178 *> \verbatim
179 *> INFO is INTEGER
180 *> = 0: successful exit.
181 *> < 0: if INFO = -i, the i-th argument had an illegal value.
182 *> > 0: if INFO = 1, a singular value did not converge
183 *> \endverbatim
184 *
185 * Authors:
186 * ========
187 *
188 *> \author Univ. of Tennessee
189 *> \author Univ. of California Berkeley
190 *> \author Univ. of Colorado Denver
191 *> \author NAG Ltd.
192 *
193 *> \date June 2016
194 *
195 *> \ingroup auxOTHERauxiliary
196 *
197 *> \par Contributors:
198 * ==================
199 *>
200 *> Ming Gu and Huan Ren, Computer Science Division, University of
201 *> California at Berkeley, USA
202 *>
203 * =====================================================================
204  SUBROUTINE slasd1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT,
205  \$ idxq, iwork, work, info )
206 *
207 * -- LAPACK auxiliary routine (version 3.6.1) --
208 * -- LAPACK is a software package provided by Univ. of Tennessee, --
209 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
210 * June 2016
211 *
212 * .. Scalar Arguments ..
213  INTEGER INFO, LDU, LDVT, NL, NR, SQRE
214  REAL ALPHA, BETA
215 * ..
216 * .. Array Arguments ..
217  INTEGER IDXQ( * ), IWORK( * )
218  REAL D( * ), U( ldu, * ), VT( ldvt, * ), WORK( * )
219 * ..
220 *
221 * =====================================================================
222 *
223 * .. Parameters ..
224 *
225  REAL ONE, ZERO
226  parameter ( one = 1.0e+0, zero = 0.0e+0 )
227 * ..
228 * .. Local Scalars ..
229  INTEGER COLTYP, I, IDX, IDXC, IDXP, IQ, ISIGMA, IU2,
230  \$ ivt2, iz, k, ldq, ldu2, ldvt2, m, n, n1, n2
231  REAL ORGNRM
232 * ..
233 * .. External Subroutines ..
234  EXTERNAL slamrg, slascl, slasd2, slasd3, xerbla
235 * ..
236 * .. Intrinsic Functions ..
237  INTRINSIC abs, max
238 * ..
239 * .. Executable Statements ..
240 *
241 * Test the input parameters.
242 *
243  info = 0
244 *
245  IF( nl.LT.1 ) THEN
246  info = -1
247  ELSE IF( nr.LT.1 ) THEN
248  info = -2
249  ELSE IF( ( sqre.LT.0 ) .OR. ( sqre.GT.1 ) ) THEN
250  info = -3
251  END IF
252  IF( info.NE.0 ) THEN
253  CALL xerbla( 'SLASD1', -info )
254  RETURN
255  END IF
256 *
257  n = nl + nr + 1
258  m = n + sqre
259 *
260 * The following values are for bookkeeping purposes only. They are
261 * integer pointers which indicate the portion of the workspace
262 * used by a particular array in SLASD2 and SLASD3.
263 *
264  ldu2 = n
265  ldvt2 = m
266 *
267  iz = 1
268  isigma = iz + m
269  iu2 = isigma + n
270  ivt2 = iu2 + ldu2*n
271  iq = ivt2 + ldvt2*m
272 *
273  idx = 1
274  idxc = idx + n
275  coltyp = idxc + n
276  idxp = coltyp + n
277 *
278 * Scale.
279 *
280  orgnrm = max( abs( alpha ), abs( beta ) )
281  d( nl+1 ) = zero
282  DO 10 i = 1, n
283  IF( abs( d( i ) ).GT.orgnrm ) THEN
284  orgnrm = abs( d( i ) )
285  END IF
286  10 CONTINUE
287  CALL slascl( 'G', 0, 0, orgnrm, one, n, 1, d, n, info )
288  alpha = alpha / orgnrm
289  beta = beta / orgnrm
290 *
291 * Deflate singular values.
292 *
293  CALL slasd2( nl, nr, sqre, k, d, work( iz ), alpha, beta, u, ldu,
294  \$ vt, ldvt, work( isigma ), work( iu2 ), ldu2,
295  \$ work( ivt2 ), ldvt2, iwork( idxp ), iwork( idx ),
296  \$ iwork( idxc ), idxq, iwork( coltyp ), info )
297 *
298 * Solve Secular Equation and update singular vectors.
299 *
300  ldq = k
301  CALL slasd3( nl, nr, sqre, k, d, work( iq ), ldq, work( isigma ),
302  \$ u, ldu, work( iu2 ), ldu2, vt, ldvt, work( ivt2 ),
303  \$ ldvt2, iwork( idxc ), iwork( coltyp ), work( iz ),
304  \$ info )
305 *
306 * Report the possible convergence failure.
307 *
308  IF( info.NE.0 ) THEN
309  RETURN
310  END IF
311 *
312 * Unscale.
313 *
314  CALL slascl( 'G', 0, 0, one, orgnrm, n, 1, d, n, info )
315 *
316 * Prepare the IDXQ sorting permutation.
317 *
318  n1 = k
319  n2 = n - k
320  CALL slamrg( n1, n2, d, 1, -1, idxq )
321 *
322  RETURN
323 *
324 * End of SLASD1
325 *
326  END
subroutine slasd1(NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT, IDXQ, IWORK, WORK, INFO)
SLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc...
Definition: slasd1.f:206
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 slasd3(NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2, LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z, INFO)
SLASD3 finds all square roots of the roots of the secular equation, as defined by the values in D and...
Definition: slasd3.f:227
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slasd2(NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT, LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX, IDXC, IDXQ, COLTYP, INFO)
SLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc...
Definition: slasd2.f:271
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