LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
dlasd1.f
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1 *> \brief \b DLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DLASD1( 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 * DOUBLE PRECISION ALPHA, BETA
27 * ..
28 * .. Array Arguments ..
29 * INTEGER IDXQ( * ), IWORK( * )
30 * DOUBLE PRECISION D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> DLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B,
40 *> where N = NL + NR + 1 and M = N + SQRE. DLASD1 is called from DLASD0.
41 *>
42 *> A related subroutine DLASD7 handles the case in which the singular
43 *> values (and the singular vectors in factored form) are desired.
44 *>
45 *> DLASD1 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 DLASD2.
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 DLASD4 (as called
70 *> by DLASD3). 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 DOUBLE PRECISION array,
107 *> dimension (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 DOUBLE PRECISION
117 *> Contains the diagonal element associated with the added row.
118 *> \endverbatim
119 *>
120 *> \param[in,out] BETA
121 *> \verbatim
122 *> BETA is DOUBLE PRECISION
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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 *> \ingroup OTHERauxiliary
194 *
195 *> \par Contributors:
196 * ==================
197 *>
198 *> Ming Gu and Huan Ren, Computer Science Division, University of
199 *> California at Berkeley, USA
200 *>
201 * =====================================================================
202  SUBROUTINE dlasd1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT,
203  $ IDXQ, IWORK, WORK, INFO )
204 *
205 * -- LAPACK auxiliary routine --
206 * -- LAPACK is a software package provided by Univ. of Tennessee, --
207 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
208 *
209 * .. Scalar Arguments ..
210  INTEGER INFO, LDU, LDVT, NL, NR, SQRE
211  DOUBLE PRECISION ALPHA, BETA
212 * ..
213 * .. Array Arguments ..
214  INTEGER IDXQ( * ), IWORK( * )
215  DOUBLE PRECISION D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
216 * ..
217 *
218 * =====================================================================
219 *
220 * .. Parameters ..
221 *
222  DOUBLE PRECISION ONE, ZERO
223  parameter( one = 1.0d+0, zero = 0.0d+0 )
224 * ..
225 * .. Local Scalars ..
226  INTEGER COLTYP, I, IDX, IDXC, IDXP, IQ, ISIGMA, IU2,
227  $ ivt2, iz, k, ldq, ldu2, ldvt2, m, n, n1, n2
228  DOUBLE PRECISION ORGNRM
229 * ..
230 * .. External Subroutines ..
231  EXTERNAL dlamrg, dlascl, dlasd2, dlasd3, xerbla
232 * ..
233 * .. Intrinsic Functions ..
234  INTRINSIC abs, max
235 * ..
236 * .. Executable Statements ..
237 *
238 * Test the input parameters.
239 *
240  info = 0
241 *
242  IF( nl.LT.1 ) THEN
243  info = -1
244  ELSE IF( nr.LT.1 ) THEN
245  info = -2
246  ELSE IF( ( sqre.LT.0 ) .OR. ( sqre.GT.1 ) ) THEN
247  info = -3
248  END IF
249  IF( info.NE.0 ) THEN
250  CALL xerbla( 'DLASD1', -info )
251  RETURN
252  END IF
253 *
254  n = nl + nr + 1
255  m = n + sqre
256 *
257 * The following values are for bookkeeping purposes only. They are
258 * integer pointers which indicate the portion of the workspace
259 * used by a particular array in DLASD2 and DLASD3.
260 *
261  ldu2 = n
262  ldvt2 = m
263 *
264  iz = 1
265  isigma = iz + m
266  iu2 = isigma + n
267  ivt2 = iu2 + ldu2*n
268  iq = ivt2 + ldvt2*m
269 *
270  idx = 1
271  idxc = idx + n
272  coltyp = idxc + n
273  idxp = coltyp + n
274 *
275 * Scale.
276 *
277  orgnrm = max( abs( alpha ), abs( beta ) )
278  d( nl+1 ) = zero
279  DO 10 i = 1, n
280  IF( abs( d( i ) ).GT.orgnrm ) THEN
281  orgnrm = abs( d( i ) )
282  END IF
283  10 CONTINUE
284  CALL dlascl( 'G', 0, 0, orgnrm, one, n, 1, d, n, info )
285  alpha = alpha / orgnrm
286  beta = beta / orgnrm
287 *
288 * Deflate singular values.
289 *
290  CALL dlasd2( nl, nr, sqre, k, d, work( iz ), alpha, beta, u, ldu,
291  $ vt, ldvt, work( isigma ), work( iu2 ), ldu2,
292  $ work( ivt2 ), ldvt2, iwork( idxp ), iwork( idx ),
293  $ iwork( idxc ), idxq, iwork( coltyp ), info )
294 *
295 * Solve Secular Equation and update singular vectors.
296 *
297  ldq = k
298  CALL dlasd3( nl, nr, sqre, k, d, work( iq ), ldq, work( isigma ),
299  $ u, ldu, work( iu2 ), ldu2, vt, ldvt, work( ivt2 ),
300  $ ldvt2, iwork( idxc ), iwork( coltyp ), work( iz ),
301  $ info )
302 *
303 * Report the convergence failure.
304 *
305  IF( info.NE.0 ) THEN
306  RETURN
307  END IF
308 *
309 * Unscale.
310 *
311  CALL dlascl( 'G', 0, 0, one, orgnrm, n, 1, d, n, info )
312 *
313 * Prepare the IDXQ sorting permutation.
314 *
315  n1 = k
316  n2 = n - k
317  CALL dlamrg( n1, n2, d, 1, -1, idxq )
318 *
319  RETURN
320 *
321 * End of DLASD1
322 *
323  END
subroutine dlasd2(NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT, LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX, IDXC, IDXQ, COLTYP, INFO)
DLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.
Definition: dlasd2.f:269
subroutine dlasd3(NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2, LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z, INFO)
DLASD3 finds all square roots of the roots of the secular equation, as defined by the values in D and...
Definition: dlasd3.f:224
subroutine dlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: dlascl.f:143
subroutine dlasd1(NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT, IDXQ, IWORK, WORK, INFO)
DLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.
Definition: dlasd1.f:204
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dlamrg(N1, N2, A, DTRD1, DTRD2, INDEX)
DLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single...
Definition: dlamrg.f:99