CLALSA(l) ) CLALSA(l)NAME
CLALSA - i an itermediate step in solving the least squares problem by
computing the SVD of the coefficient matrix in compact form (The singu‐
lar vectors are computed as products of simple orthorgonal matrices.)
SYNOPSIS
SUBROUTINE CLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U, LDU,
VT, K, DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL,
PERM, GIVNUM, C, S, RWORK, IWORK, INFO )
INTEGER ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS, SML‐
SIZ
INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), K( *
), PERM( LDGCOL, * )
REAL C( * ), DIFL( LDU, * ), DIFR( LDU, * ), GIVNUM( LDU,
* ), POLES( LDU, * ), RWORK( * ), S( * ), U( LDU, *
), VT( LDU, * ), Z( LDU, * )
COMPLEX B( LDB, * ), BX( LDBX, * )
PURPOSE
CLALSA is an itermediate step in solving the least squares problem by
computing the SVD of the coefficient matrix in compact form (The singu‐
lar vectors are computed as products of simple orthorgonal matrices.).
If ICOMPQ = 0, CLALSA applies the inverse of the left singular vector
matrix of an upper bidiagonal matrix to the right hand side; and if
ICOMPQ = 1, CLALSA applies the right singular vector matrix to the
right hand side. The singular vector matrices were generated in compact
form by CLALSA.
ARGUMENTS
ICOMPQ (input) INTEGER Specifies whether the left or the right singular
vector matrix is involved. = 0: Left singular vector matrix
= 1: Right singular vector matrix
SMLSIZ (input) INTEGER The maximum size of the subproblems at the bot‐
tom of the computation tree.
N (input) INTEGER
The row and column dimensions of the upper bidiagonal matrix.
NRHS (input) INTEGER
The number of columns of B and BX. NRHS must be at least 1.
B (input) 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 in the calling subprogram. LDB must
be at least max(1,MAX( M, N ) ).
BX (output) COMPLEX array, dimension ( LDBX, NRHS )
On exit, the result of applying the left or right singular vec‐
tor matrix to B.
LDBX (input) INTEGER
The leading dimension of BX.
U (input) REAL array, dimension ( LDU, SMLSIZ ).
On entry, U contains the left singular vector matrices of all
subproblems at the bottom level.
LDU (input) INTEGER, LDU = > N.
The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
GIVNUM, and Z.
VT (input) REAL array, dimension ( LDU, SMLSIZ+1 ).
On entry, VT' contains the right singular vector matrices of all
subproblems at the bottom level.
K (input) INTEGER array, dimension ( N ).
DIFL (input) REAL array, dimension ( LDU, NLVL ).
where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
DIFR (input) REAL array, dimension ( LDU, 2 * NLVL ).
On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record distances
between singular values on the I-th level and singular values on
the (I -1)-th level, and DIFR(*, 2 * I) record the normalizing
factors of the right singular vectors matrices of subproblems on
I-th level.
Z (input) REAL array, dimension ( LDU, NLVL ).
On entry, Z(1, I) contains the components of the deflation-
adjusted updating row vector for subproblems on the I-th level.
POLES (input) REAL array, dimension ( LDU, 2 * NLVL ).
On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
singular values involved in the secular equations on the I-th
level.
GIVPTR (input) INTEGER array, dimension ( N ). On entry,
GIVPTR( I ) records the number of Givens rotations performed on
the I-th problem on the computation tree.
GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the
locations of Givens rotations performed on the I-th level on the
computation tree.
LDGCOL (input) INTEGER, LDGCOL = > N. The leading dimension of
arrays GIVCOL and PERM.
PERM (input) INTEGER array, dimension ( LDGCOL, NLVL ).
On entry, PERM(*, I) records permutations done on the I-th level
of the computation tree.
GIVNUM (input) REAL array, dimension ( LDU, 2 * NLVL ). On
entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S- values
of Givens rotations performed on the I-th level on the computa‐
tion tree.
C (input) REAL array, dimension ( N ).
On entry, if the I-th subproblem is not square, C( I ) contains
the C-value of a Givens rotation related to the right null space
of the I-th subproblem.
S (input) REAL array, dimension ( N ).
On entry, if the I-th subproblem is not square, S( I ) contains
the S-value of a Givens rotation related to the right null space
of the I-th subproblem.
RWORK (workspace) REAL array, dimension at least
max ( N, (SMLSZ+1)*NRHS*3 ).
IWORK (workspace) INTEGER array.
The dimension must be at least 3 * N
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 version 3.0 15 June 2000 CLALSA(l)