ZLALSA(1) LAPACK routine (version 3.2) ZLALSA(1)NAME
ZLALSA - 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.)
SYNOPSIS
SUBROUTINE ZLALSA( 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, * )
DOUBLE PRECISION C( * ), DIFL( LDU, * ), DIFR( LDU, * ),
GIVNUM( LDU, * ), POLES( LDU, * ), RWORK( * ), S( *
), U( LDU, * ), VT( LDU, * ), Z( LDU, * )
COMPLEX*16 B( LDB, * ), BX( LDBX, * )
PURPOSE
ZLALSA 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, ZLALSA applies the inverse of the left singular vector
matrix of an upper bidiagonal matrix to the right hand side; and if
ICOMPQ = 1, ZLALSA applies the right singular vector matrix to the
right hand side. The singular vector matrices were generated in compact
form by ZLALSA.
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 bottom 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/output) COMPLEX*16 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*16 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension ( LDU, NLVL ).
where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
DIFR (input) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 per‐
formed 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 loca‐
tions 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) DOUBLE PRECISION 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 computation tree.
C (input) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 routine (version 3.2) November 2008 ZLALSA(1)
