1DLASD2(1) LAPACK auxiliary routine (version 3.2) DLASD2(1)
2
6 DLASD2 - merges the two sets of singular values together into a single
7 sorted set
8
10 SUBROUTINE DLASD2( NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT,
11 LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX, IDXC,
12 IDXQ, COLTYP, INFO )
13 INTEGER INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE
15 DOUBLE PRECISION ALPHA, BETA
17 INTEGER COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ), IDXQ( *
19 )
20 DOUBLE PRECISION D( * ), DSIGMA( * ), U( LDU, * ), U2(
22 LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ), Z( * )
23
25 DLASD2 merges the two sets of singular values together into a single
26 sorted set. Then it tries to deflate the size of the problem. There
27 are two ways in which deflation can occur: when two or more singular
28 values are close together or if there is a tiny entry in the Z vector.
29 For each such occurrence the order of the related secular equation
30 problem is reduced by one.
31 DLASD2 is called from DLASD1.
32
34 NL (input) INTEGER
35 The row dimension of the upper block. NL >= 1.
36 NR (input) INTEGER
38 The row dimension of the lower block. NR >= 1.
39 SQRE (input) INTEGER
41 = 0: the lower block is an NR-by-NR square matrix.
42 = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The
43 bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N
44 columns.
45 K (output) INTEGER
47 Contains the dimension of the non-deflated matrix, This is the
48 order of the related secular equation. 1 <= K <=N.
49 D (input/output) DOUBLE PRECISION array, dimension(N)
51 On entry D contains the singular values of the two submatrices
52 to be combined. On exit D contains the trailing (N-K) updated
53 singular values (those which were deflated) sorted into increas‐
54 ing order.
55 Z (output) DOUBLE PRECISION array, dimension(N)
57 On exit Z contains the updating row vector in the secular equa‐
58 tion.
59 ALPHA (input) DOUBLE PRECISION
61 Contains the diagonal element associated with the added row.
62 BETA (input) DOUBLE PRECISION
64 Contains the off-diagonal element associated with the added row.
65 U (input/output) DOUBLE PRECISION array, dimension(LDU,N)
67 On entry U contains the left singular vectors of two submatrices
68 in the two square blocks with corners at (1,1), (NL, NL), and
69 (NL+2, NL+2), (N,N). On exit U contains the trailing (N-K)
70 updated left singular vectors (those which were deflated) in its
71 last N-K columns.
72 LDU (input) INTEGER
74 The leading dimension of the array U. LDU >= N.
75 VT (input/output) DOUBLE PRECISION array, dimension(LDVT,M)
77 On entry VT' contains the right singular vectors of two subma‐
78 trices in the two square blocks with corners at (1,1), (NL+1,
79 NL+1), and (NL+2, NL+2), (M,M). On exit VT' contains the trail‐
80 ing (N-K) updated right singular vectors (those which were
81 deflated) in its last N-K columns. In case SQRE =1, the last
82 row of VT spans the right null space.
83 LDVT (input) INTEGER
85 The leading dimension of the array VT. LDVT >= M. DSIGMA (out‐
86 put) DOUBLE PRECISION array, dimension (N) Contains a copy of
87 the diagonal elements (K-1 singular values and one zero) in the
88 secular equation.
89 U2 (output) DOUBLE PRECISION array, dimension(LDU2,N)
91 Contains a copy of the first K-1 left singular vectors which
92 will be used by DLASD3 in a matrix multiply (DGEMM) to solve for
93 the new left singular vectors. U2 is arranged into four blocks.
94 The first block contains a column with 1 at NL+1 and zero every‐
95 where else; the second block contains non-zero entries only at
96 and above NL; the third contains non-zero entries only below
97 NL+1; and the fourth is dense.
98 LDU2 (input) INTEGER
100 The leading dimension of the array U2. LDU2 >= N.
101 VT2 (output) DOUBLE PRECISION array, dimension(LDVT2,N)
103 VT2' contains a copy of the first K right singular vectors which
104 will be used by DLASD3 in a matrix multiply (DGEMM) to solve for
105 the new right singular vectors. VT2 is arranged into three
106 blocks. The first block contains a row that corresponds to the
107 special 0 diagonal element in SIGMA; the second block contains
108 non-zeros only at and before NL +1; the third block contains
109 non-zeros only at and after NL +2.
110 LDVT2 (input) INTEGER
112 The leading dimension of the array VT2. LDVT2 >= M.
113 IDXP (workspace) INTEGER array dimension(N)
115 This will contain the permutation used to place deflated values
116 of D at the end of the array. On output IDXP(2:K)
117 points to the nondeflated D-values and IDXP(K+1:N) points to the
118 deflated singular values.
119 IDX (workspace) INTEGER array dimension(N)
121 This will contain the permutation used to sort the contents of D
122 into ascending order.
123 IDXC (output) INTEGER array dimension(N)
125 This will contain the permutation used to arrange the columns of
126 the deflated U matrix into three groups: the first group con‐
127 tains non-zero entries only at and above NL, the second contains
128 non-zero entries only below NL+2, and the third is dense.
129 IDXQ (input/output) INTEGER array dimension(N)
131 This contains the permutation which separately sorts the two
132 sub-problems in D into ascending order. Note that entries in
133 the first hlaf of this permutation must first be moved one posi‐
134 tion backward; and entries in the second half must first have
135 NL+1 added to their values. COLTYP (workspace/output) INTEGER
136 array dimension(N) As workspace, this will contain a label which
137 will indicate which of the following types a column in the U2
138 matrix or a row in the VT2 matrix is:
139 1 : non-zero in the upper half only
140 2 : non-zero in the lower half only
141 3 : dense
142 4 : deflated On exit, it is an array of dimension 4, with
143 COLTYP(I) being the dimension of the I-th type columns.
144 INFO (output) INTEGER
146 = 0: successful exit.
147 < 0: if INFO = -i, the i-th argument had an illegal value.
148
150 Based on contributions by
151 Ming Gu and Huan Ren, Computer Science Division, University of
152 California at Berkeley, USA
153 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 DLASD2(1)