CLALS0(1) LAPACK routine (version 3.2) CLALS0(1)NAME
CLALS0 - applies back the multiplying factors of either the left or the
right singular vector matrix of a diagonal matrix appended by a row to
the right hand side matrix B in solving the least squares problem using
the divide-and-conquer SVD approach
SYNOPSIS
SUBROUTINE CLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX, PERM,
GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL,
DIFR, Z, K, C, S, RWORK, INFO )
INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL, LDGNUM,
NL, NR, NRHS, SQRE
REAL C, S
INTEGER GIVCOL( LDGCOL, * ), PERM( * )
REAL DIFL( * ), DIFR( LDGNUM, * ), GIVNUM( LDGNUM, * ),
POLES( LDGNUM, * ), RWORK( * ), Z( * )
COMPLEX B( LDB, * ), BX( LDBX, * )
PURPOSE
CLALS0 applies back the multiplying factors of either the left or the
right singular vector matrix of a diagonal matrix appended by a row to
the right hand side matrix B in solving the least squares problem using
the divide-and-conquer SVD approach. For the left singular vector
matrix, three types of orthogonal matrices are involved:
(1L) Givens rotations: the number of such rotations is GIVPTR; the
pairs of columns/rows they were applied to are stored in GIVCOL;
and the C- and S-values of these rotations are stored in GIVNUM.
(2L) Permutation. The (NL+1)-st row of B is to be moved to the first
row, and for J=2:N, PERM(J)-th row of B is to be moved to the
J-th row.
(3L) The left singular vector matrix of the remaining matrix. For the
right singular vector matrix, four types of orthogonal matrices are
involved:
(1R) The right singular vector matrix of the remaining matrix. (2R) If
SQRE = 1, one extra Givens rotation to generate the right
null space.
(3R) The inverse transformation of (2L).
(4R) The inverse transformation of (1L).
ARGUMENTS
ICOMPQ (input) INTEGER Specifies whether singular vectors are to be
computed in factored form:
= 0: Left singular vector matrix.
= 1: Right singular vector matrix.
NL (input) INTEGER
The row dimension of the upper block. NL >= 1.
NR (input) INTEGER
The row dimension of the lower block. NR >= 1.
SQRE (input) INTEGER
= 0: the lower block is an NR-by-NR square matrix.
= 1: the lower block is an NR-by-(NR+1) rectangular matrix. The
bidiagonal matrix has row dimension N = NL + NR + 1, and column
dimension M = N + SQRE.
NRHS (input) INTEGER
The number of columns of B and BX. NRHS must be at least 1.
B (input/output) COMPLEX array, dimension ( LDB, NRHS )
On input, B contains the right hand sides of the least squares
problem in rows 1 through M. On output, B contains the solution
X in rows 1 through N.
LDB (input) INTEGER
The leading dimension of B. LDB must be at least max(1,MAX( M, N
) ).
BX (workspace) COMPLEX array, dimension ( LDBX, NRHS )
LDBX (input) INTEGER
The leading dimension of BX.
PERM (input) INTEGER array, dimension ( N )
The permutations (from deflation and sorting) applied to the two
blocks. GIVPTR (input) INTEGER The number of Givens rotations
which took place in this subproblem. GIVCOL (input) INTEGER
array, dimension ( LDGCOL, 2 ) Each pair of numbers indicates a
pair of rows/columns involved in a Givens rotation. LDGCOL
(input) INTEGER The leading dimension of GIVCOL, must be at
least N. GIVNUM (input) REAL array, dimension ( LDGNUM, 2 )
Each number indicates the C or S value used in the corresponding
Givens rotation. LDGNUM (input) INTEGER The leading dimension
of arrays DIFR, POLES and GIVNUM, must be at least K.
POLES (input) REAL array, dimension ( LDGNUM, 2 )
On entry, POLES(1:K, 1) contains the new singular values
obtained from solving the secular equation, and POLES(1:K, 2) is
an array containing the poles in the secular equation.
DIFL (input) REAL array, dimension ( K ).
On entry, DIFL(I) is the distance between I-th updated (unde‐
flated) singular value and the I-th (undeflated) old singular
value.
DIFR (input) REAL array, dimension ( LDGNUM, 2 ).
On entry, DIFR(I, 1) contains the distances between I-th updated
(undeflated) singular value and the I+1-th (undeflated) old sin‐
gular value. And DIFR(I, 2) is the normalizing factor for the I-
th right singular vector.
Z (input) REAL array, dimension ( K )
Contain the components of the deflation-adjusted updating row
vector.
K (input) INTEGER
Contains the dimension of the non-deflated matrix, This is the
order of the related secular equation. 1 <= K <=N.
C (input) REAL
C contains garbage if SQRE =0 and the C-value of a Givens rota‐
tion related to the right null space if SQRE = 1.
S (input) REAL
S contains garbage if SQRE =0 and the S-value of a Givens rota‐
tion related to the right null space if SQRE = 1.
RWORK (workspace) REAL array, dimension
( K*(1+NRHS) + 2*NRHS )
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
FURTHER DETAILS
Based on contributions by
Ming Gu and Ren-Cang Li, Computer Science Division, University of
California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA
LAPACK routine (version 3.2) November 2008 CLALS0(1)
