DLALS0(3)

NAME
DLALS0 - 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 DLALS0(
    ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO )

INTEGER
    GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL, LDGNUM, NL, NR, NRHS, SQRE

DOUBLE
    PRECISION C, S

INTEGER
    GIVCOL( LDGCOL, * ), PERM( * )

DOUBLE
    PRECISION B( LDB, * ), BX( LDBX, * ), DIFL( * ), DIFR( LDGNUM, * ), GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ), WORK( * ), Z( * )

PURPOSE
DLALS0 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) 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. LDB must be at least max(1,MAX( M, N ) ).

BX     (workspace) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension ( K ).
    On entry, DIFL(I) is the distance between I-th updated (undeflated) singular value and the I-th (undeflated) old singular value.

DIFR   (input) DOUBLE PRECISION 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 singular value. And DIFR(I, 2) is the normalizing factor for the I-th right singular vector.

Z      (input) DOUBLE PRECISION 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) DOUBLE PRECISION
    C contains garbage if SQRE =0 and the C-value of a Givens rotation related to the right null space if SQRE = 1.

S      (input) DOUBLE PRECISION
    S contains garbage if SQRE =0 and the S-value of a Givens rotation related to the right null space if SQRE = 1.

WORK   (workspace) DOUBLE PRECISION array, dimension ( K )

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