1DLALS0(1) LAPACK routine (version 3.2) DLALS0(1)
2
6 DLALS0 - applies back the multiplying factors of either the left or the
7 right singular vector matrix of a diagonal matrix appended by a row to
8 the right hand side matrix B in solving the least squares problem using
9 the divide-and-conquer SVD approach
10
12 SUBROUTINE DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX, PERM,
13 GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL,
14 DIFR, Z, K, C, S, WORK, INFO )
15 INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL, LDGNUM,
17 NL, NR, NRHS, SQRE
18 DOUBLE PRECISION C, S
20 INTEGER GIVCOL( LDGCOL, * ), PERM( * )
22 DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ), DIFL( * ),
24 DIFR( LDGNUM, * ), GIVNUM( LDGNUM, * ), POLES(
25 LDGNUM, * ), WORK( * ), Z( * )
26
28 DLALS0 applies back the multiplying factors of either the left or the
29 right singular vector matrix of a diagonal matrix appended by a row to
30 the right hand side matrix B in solving the least squares problem using
31 the divide-and-conquer SVD approach. For the left singular vector
32 matrix, three types of orthogonal matrices are involved:
33 (1L) Givens rotations: the number of such rotations is GIVPTR; the
34 pairs of columns/rows they were applied to are stored in GIVCOL;
35 and the C- and S-values of these rotations are stored in GIVNUM.
36 (2L) Permutation. The (NL+1)-st row of B is to be moved to the first
37 row, and for J=2:N, PERM(J)-th row of B is to be moved to the
38 J-th row.
39 (3L) The left singular vector matrix of the remaining matrix. For the
40 right singular vector matrix, four types of orthogonal matrices are
41 involved:
42 (1R) The right singular vector matrix of the remaining matrix. (2R) If
43 SQRE = 1, one extra Givens rotation to generate the right
44 null space.
45 (3R) The inverse transformation of (2L).
46 (4R) The inverse transformation of (1L).
47
49 ICOMPQ (input) INTEGER Specifies whether singular vectors are to be
50 computed in factored form:
51 = 0: Left singular vector matrix.
52 = 1: Right singular vector matrix.
53 NL (input) INTEGER
55 The row dimension of the upper block. NL >= 1.
56 NR (input) INTEGER
58 The row dimension of the lower block. NR >= 1.
59 SQRE (input) INTEGER
61 = 0: the lower block is an NR-by-NR square matrix.
62 = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The
63 bidiagonal matrix has row dimension N = NL + NR + 1, and column
64 dimension M = N + SQRE.
65 NRHS (input) INTEGER
67 The number of columns of B and BX. NRHS must be at least 1.
68 B (input/output) DOUBLE PRECISION array, dimension ( LDB, NRHS )
70 On input, B contains the right hand sides of the least squares
71 problem in rows 1 through M. On output, B contains the solution
72 X in rows 1 through N.
73 LDB (input) INTEGER
75 The leading dimension of B. LDB must be at least max(1,MAX( M, N
76 ) ).
77 BX (workspace) DOUBLE PRECISION array, dimension ( LDBX, NRHS )
79 LDBX (input) INTEGER
81 The leading dimension of BX.
82 PERM (input) INTEGER array, dimension ( N )
84 The permutations (from deflation and sorting) applied to the two
85 blocks. GIVPTR (input) INTEGER The number of Givens rotations
86 which took place in this subproblem. GIVCOL (input) INTEGER
87 array, dimension ( LDGCOL, 2 ) Each pair of numbers indicates a
88 pair of rows/columns involved in a Givens rotation. LDGCOL
89 (input) INTEGER The leading dimension of GIVCOL, must be at
90 least N. GIVNUM (input) DOUBLE PRECISION array, dimension (
91 LDGNUM, 2 ) Each number indicates the C or S value used in the
92 corresponding Givens rotation. LDGNUM (input) INTEGER The lead‐
93 ing dimension of arrays DIFR, POLES and GIVNUM, must be at least
94 K.
95 POLES (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
97 On entry, POLES(1:K, 1) contains the new singular values
98 obtained from solving the secular equation, and POLES(1:K, 2) is
99 an array containing the poles in the secular equation.
100 DIFL (input) DOUBLE PRECISION array, dimension ( K ).
102 On entry, DIFL(I) is the distance between I-th updated (unde‐
103 flated) singular value and the I-th (undeflated) old singular
104 value.
105 DIFR (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ).
107 On entry, DIFR(I, 1) contains the distances between I-th updated
108 (undeflated) singular value and the I+1-th (undeflated) old sin‐
109 gular value. And DIFR(I, 2) is the normalizing factor for the I-
110 th right singular vector.
111 Z (input) DOUBLE PRECISION array, dimension ( K )
113 Contain the components of the deflation-adjusted updating row
114 vector.
115 K (input) INTEGER
117 Contains the dimension of the non-deflated matrix, This is the
118 order of the related secular equation. 1 <= K <=N.
119 C (input) DOUBLE PRECISION
121 C contains garbage if SQRE =0 and the C-value of a Givens rota‐
122 tion related to the right null space if SQRE = 1.
123 S (input) DOUBLE PRECISION
125 S contains garbage if SQRE =0 and the S-value of a Givens rota‐
126 tion related to the right null space if SQRE = 1.
127 WORK (workspace) DOUBLE PRECISION array, dimension ( K )
129 INFO (output) INTEGER
131 = 0: successful exit.
132 < 0: if INFO = -i, the i-th argument had an illegal value.
133
135 Based on contributions by
136 Ming Gu and Ren-Cang Li, Computer Science Division, University of
137 California at Berkeley, USA
138 Osni Marques, LBNL/NERSC, USA
139 LAPACK routine (version 3.2) November 2008 DLALS0(1)