CLALS0 - applie 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

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, * )

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).

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 (undeflated) 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 singular 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 rotation 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 rotation 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.

Based on contributions by

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

Osni Marques, LBNL/NERSC, USA