1SLASD6(1) LAPACK auxiliary routine (version 3.2) SLASD6(1)
2
6 SLASD6 - computes the SVD of an updated upper bidiagonal matrix B
7 obtained by merging two smaller ones by appending a row
8
10 SUBROUTINE SLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA, IDXQ,
11 PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES,
12 DIFL, DIFR, Z, K, C, S, WORK, IWORK, INFO )
13 INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL, NR,
15 SQRE
16 REAL ALPHA, BETA, C, S
18 INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ), PERM( *
20 )
21 REAL D( * ), DIFL( * ), DIFR( * ), GIVNUM( LDGNUM, * ),
23 POLES( LDGNUM, * ), VF( * ), VL( * ), WORK( * ), Z(
24 * )
25
27 SLASD6 computes the SVD of an updated upper bidiagonal matrix B
28 obtained by merging two smaller ones by appending a row. This routine
29 is used only for the problem which requires all singular values and
30 optionally singular vector matrices in factored form. B is an N-by-M
31 matrix with N = NL + NR + 1 and M = N + SQRE. A related subroutine,
32 SLASD1, handles the case in which all singular values and singular vec‐
33 tors of the bidiagonal matrix are desired. SLASD6 computes the SVD as
34 follows:
35 ( D1(in) 0 0 0 )
36 B = U(in) * ( Z1' a Z2' b ) * VT(in)
37 ( 0 0 D2(in) 0 )
38 = U(out) * ( D(out) 0) * VT(out)
39 where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M
40 with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros else‐
41 where; and the entry b is empty if SQRE = 0.
42 The singular values of B can be computed using D1, D2, the first compo‐
43 nents of all the right singular vectors of the lower block, and the
44 last components of all the right singular vectors of the upper block.
45 These components are stored and updated in VF and VL, respectively, in
46 SLASD6. Hence U and VT are not explicitly referenced.
47 The singular values are stored in D. The algorithm consists of two
48 stages:
49 The first stage consists of deflating the size of the problem
50 when there are multiple singular values or if there is a zero
51 in the Z vector. For each such occurence the dimension of the
52 secular equation problem is reduced by one. This stage is
53 performed by the routine SLASD7.
54 The second stage consists of calculating the updated
55 singular values. This is done by finding the roots of the
56 secular equation via the routine SLASD4 (as called by SLASD8).
57 This routine also updates VF and VL and computes the distances
58 between the updated singular values and the old singular
59 values.
60 SLASD6 is called from SLASDA.
61
63 ICOMPQ (input) INTEGER Specifies whether singular vectors are to be
64 computed in factored form:
65 = 0: Compute singular values only.
66 = 1: Compute singular vectors in factored form as well.
67 NL (input) INTEGER
69 The row dimension of the upper block. NL >= 1.
70 NR (input) INTEGER
72 The row dimension of the lower block. NR >= 1.
73 SQRE (input) INTEGER
75 = 0: the lower block is an NR-by-NR square matrix.
76 = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The
77 bidiagonal matrix has row dimension N = NL + NR + 1, and column
78 dimension M = N + SQRE.
79 D (input/output) REAL array, dimension (NL+NR+1).
81 On entry D(1:NL,1:NL) contains the singular values of the
82 upper block, and D(NL+2:N) contains the singular values
83 of the lower block. On exit D(1:N) contains the singular values
84 of the modified matrix.
85 VF (input/output) REAL array, dimension (M)
87 On entry, VF(1:NL+1) contains the first components of all
88 right singular vectors of the upper block; and VF(NL+2:M) con‐
89 tains the first components of all right singular vectors of the
90 lower block. On exit, VF contains the first components of all
91 right singular vectors of the bidiagonal matrix.
92 VL (input/output) REAL array, dimension (M)
94 On entry, VL(1:NL+1) contains the last components of all
95 right singular vectors of the upper block; and VL(NL+2:M) con‐
96 tains the last components of all right singular vectors of the
97 lower block. On exit, VL contains the last components of all
98 right singular vectors of the bidiagonal matrix.
99 ALPHA (input/output) REAL
101 Contains the diagonal element associated with the added row.
102 BETA (input/output) REAL
104 Contains the off-diagonal element associated with the added row.
105 IDXQ (output) INTEGER array, dimension (N)
107 This contains the permutation which will reintegrate the sub‐
108 problem just solved back into sorted order, i.e. D( IDXQ( I =
109 1, N ) ) will be in ascending order.
110 PERM (output) INTEGER array, dimension ( N )
112 The permutations (from deflation and sorting) to be applied to
113 each block. Not referenced if ICOMPQ = 0. GIVPTR (output) INTE‐
114 GER The number of Givens rotations which took place in this sub‐
115 problem. Not referenced if ICOMPQ = 0. GIVCOL (output) INTEGER
116 array, dimension ( LDGCOL, 2 ) Each pair of numbers indicates a
117 pair of columns to take place in a Givens rotation. Not refer‐
118 enced if ICOMPQ = 0. LDGCOL (input) INTEGER leading dimension
119 of GIVCOL, must be at least N. GIVNUM (output) REAL array,
120 dimension ( LDGNUM, 2 ) Each number indicates the C or S value
121 to be used in the corresponding Givens rotation. Not referenced
122 if ICOMPQ = 0. LDGNUM (input) INTEGER The leading dimension of
123 GIVNUM and POLES, must be at least N.
124 POLES (output) REAL array, dimension ( LDGNUM, 2 )
126 On exit, POLES(1,*) is an array containing the new singular val‐
127 ues obtained from solving the secular equation, and POLES(2,*)
128 is an array containing the poles in the secular equation. Not
129 referenced if ICOMPQ = 0.
130 DIFL (output) REAL array, dimension ( N )
132 On exit, DIFL(I) is the distance between I-th updated (unde‐
133 flated) singular value and the I-th (undeflated) old singular
134 value.
135 DIFR (output) REAL array,
137 dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and dimension ( N ) if
138 ICOMPQ = 0. On exit, DIFR(I, 1) is the distance between I-th
139 updated (undeflated) singular value and the I+1-th (undeflated)
140 old singular value. If ICOMPQ = 1, DIFR(1:K,2) is an array con‐
141 taining the normalizing factors for the right singular vector
142 matrix. See SLASD8 for details on DIFL and DIFR.
143 Z (output) REAL array, dimension ( M )
145 The first elements of this array contain the components of the
146 deflation-adjusted updating row vector.
147 K (output) INTEGER
149 Contains the dimension of the non-deflated matrix, This is the
150 order of the related secular equation. 1 <= K <=N.
151 C (output) REAL
153 C contains garbage if SQRE =0 and the C-value of a Givens rota‐
154 tion related to the right null space if SQRE = 1.
155 S (output) REAL
157 S contains garbage if SQRE =0 and the S-value of a Givens rota‐
158 tion related to the right null space if SQRE = 1.
159 WORK (workspace) REAL array, dimension ( 4 * M )
161 IWORK (workspace) INTEGER array, dimension ( 3 * N )
163 INFO (output) INTEGER
165 = 0: successful exit.
166 < 0: if INFO = -i, the i-th argument had an illegal value.
167 > 0: if INFO = 1, an singular value did not converge
168
170 Based on contributions by
171 Ming Gu and Huan Ren, Computer Science Division, University of
172 California at Berkeley, USA
173 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 SLASD6(1)