 DLALSA(3) is an itermediate step in solving the least squares problem by computing the SVD of the coefficient matrix in compact form (The singular vectors are computed as products of simple orthorgonal matrices.)

## SYNOPSIS

SUBROUTINE DLALSA(
ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U, LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK, IWORK, INFO )

INTEGER ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS, SMLSIZ

INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), K( * ), PERM( LDGCOL, * )

DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ), C( * ), DIFL( LDU, * ), DIFR( LDU, * ), GIVNUM( LDU, * ), POLES( LDU, * ), S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ), Z( LDU, * )

## PURPOSE

DLALSA is an itermediate step in solving the least squares problem by computing the SVD of the coefficient matrix in compact form (The singular vectors are computed as products of simple orthorgonal matrices.). If ICOMPQ = 0, DLALSA applies the inverse of the left singular vector matrix of an upper bidiagonal matrix to the right hand side; and if ICOMPQ = 1, DLALSA applies the right singular vector matrix to the right hand side. The singular vector matrices were generated in compact form by DLALSA.

## 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension ( LDBX, NRHS )
On exit, the result of applying the left or right singular vector matrix to B.
LDBX (input) INTEGER
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 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) 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.
WORK (workspace) DOUBLE PRECISION array.
The dimension must be at least N.
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