1ZLALS0(1) LAPACK routine (version 3.1) ZLALS0(1)
2
6 ZLALS0 - back the multiplying factors of either the left or the right
7 singular vector matrix of a diagonal matrix appended by a row to the
8 right hand side matrix B in solving the least squares problem using the
9 divide-and-conquer SVD approach
10
12 SUBROUTINE ZLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX, PERM,
13 GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL,
14 DIFR, Z, K, C, S, RWORK, 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 DIFL( * ), DIFR( LDGNUM, * ), GIVNUM(
24 LDGNUM, * ), POLES( LDGNUM, * ), RWORK( * ), Z( * )
25 COMPLEX*16 B( LDB, * ), BX( LDBX, * )
27
29 ZLALS0 applies back the multiplying factors of either the left or the
30 right singular vector matrix of a diagonal matrix appended by a row to
31 the right hand side matrix B in solving the least squares problem using
32 the divide-and-conquer SVD approach.
33 For the left singular vector matrix, three types of orthogonal matrices
35 are involved:
36 (1L) Givens rotations: the number of such rotations is GIVPTR; the
38 pairs of columns/rows they were applied to are stored in GIVCOL;
39 and the C- and S-values of these rotations are stored in GIVNUM.
40 (2L) Permutation. The (NL+1)-st row of B is to be moved to the first
42 row, and for J=2:N, PERM(J)-th row of B is to be moved to the
43 J-th row.
44 (3L) The left singular vector matrix of the remaining matrix.
46 For the right singular vector matrix, four types of orthogonal matrices
48 are involved:
49 (1R) The right singular vector matrix of the remaining matrix.
51 (2R) If SQRE = 1, one extra Givens rotation to generate the right
53 null space.
54 (3R) The inverse transformation of (2L).
56 (4R) The inverse transformation of (1L).
58
61 ICOMPQ (input) INTEGER Specifies whether singular vectors are to be
62 computed in factored form:
63 = 0: Left singular vector matrix.
64 = 1: Right singular vector matrix.
65 NL (input) INTEGER
67 The row dimension of the upper block. NL >= 1.
68 NR (input) INTEGER
70 The row dimension of the lower block. NR >= 1.
71 SQRE (input) INTEGER
73 = 0: the lower block is an NR-by-NR square matrix.
74 = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
75 The bidiagonal matrix has row dimension N = NL + NR + 1, and
77 column dimension M = N + SQRE.
78 NRHS (input) INTEGER
80 The number of columns of B and BX. NRHS must be at least 1.
81 B (input/output) COMPLEX*16 array, dimension ( LDB, NRHS )
83 On input, B contains the right hand sides of the least squares
84 problem in rows 1 through M. On output, B contains the solution
85 X in rows 1 through N.
86 LDB (input) INTEGER
88 The leading dimension of B. LDB must be at least max(1,MAX( M, N
89 ) ).
90 BX (workspace) COMPLEX*16 array, dimension ( LDBX, NRHS )
92 LDBX (input) INTEGER
94 The leading dimension of BX.
95 PERM (input) INTEGER array, dimension ( N )
97 The permutations (from deflation and sorting) applied to the two
98 blocks.
99 GIVPTR (input) INTEGER The number of Givens rotations which took
101 place in this subproblem.
102 GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 ) Each pair
104 of numbers indicates a pair of rows/columns involved in a Givens
105 rotation.
106 LDGCOL (input) INTEGER The leading dimension of GIVCOL, must be
108 at least N.
109 GIVNUM (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
111 Each number indicates the C or S value used in the corresponding
112 Givens rotation.
113 LDGNUM (input) INTEGER The leading dimension of arrays DIFR,
115 POLES and GIVNUM, must be at least K.
116 POLES (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
118 On entry, POLES(1:K, 1) contains the new singular values
119 obtained from solving the secular equation, and POLES(1:K, 2) is
120 an array containing the poles in the secular equation.
121 DIFL (input) DOUBLE PRECISION array, dimension ( K ).
123 On entry, DIFL(I) is the distance between I-th updated (unde‐
124 flated) singular value and the I-th (undeflated) old singular
125 value.
126 DIFR (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ).
128 On entry, DIFR(I, 1) contains the distances between I-th updated
129 (undeflated) singular value and the I+1-th (undeflated) old sin‐
130 gular value. And DIFR(I, 2) is the normalizing factor for the I-
131 th right singular vector.
132 Z (input) DOUBLE PRECISION array, dimension ( K )
134 Contain the components of the deflation-adjusted updating row
135 vector.
136 K (input) INTEGER
138 Contains the dimension of the non-deflated matrix, This is the
139 order of the related secular equation. 1 <= K <=N.
140 C (input) DOUBLE PRECISION
142 C contains garbage if SQRE =0 and the C-value of a Givens rota‐
143 tion related to the right null space if SQRE = 1.
144 S (input) DOUBLE PRECISION
146 S contains garbage if SQRE =0 and the S-value of a Givens rota‐
147 tion related to the right null space if SQRE = 1.
148 RWORK (workspace) DOUBLE PRECISION array, dimension
150 ( K*(1+NRHS) + 2*NRHS )
151 INFO (output) INTEGER
153 = 0: successful exit.
154 < 0: if INFO = -i, the i-th argument had an illegal value.
155
157 Based on contributions by
158 Ming Gu and Ren-Cang Li, Computer Science Division, University of
159 California at Berkeley, USA
160 Osni Marques, LBNL/NERSC, USA
161 LAPACK routine (version 3.1) November 2006 ZLALS0(1)