!>
!> 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

!>          ICOMPQ is INTEGER
!>         Specifies whether singular vectors are to be computed in
!>         factored form:
!>         = 0: Left singular vector matrix.
!>         = 1: Right singular vector matrix.
!> 

NL

!>          NL is INTEGER
!>         The row dimension of the upper block. NL >= 1.
!> 

NR

!>          NR is INTEGER
!>         The row dimension of the lower block. NR >= 1.
!> 

SQRE

!>          SQRE is 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

!>          NRHS is INTEGER
!>         The number of columns of B and BX. NRHS must be at least 1.
!> 

B

!>          B is 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

!>          LDB is INTEGER
!>         The leading dimension of B. LDB must be at least
!>         max(1,MAX( M, N ) ).
!> 

BX

!>          BX is COMPLEX array, dimension ( LDBX, NRHS )
!> 

LDBX

!>          LDBX is INTEGER
!>         The leading dimension of BX.
!> 

PERM

!>          PERM is INTEGER array, dimension ( N )
!>         The permutations (from deflation and sorting) applied
!>         to the two blocks.
!> 

GIVPTR

!>          GIVPTR is INTEGER
!>         The number of Givens rotations which took place in this
!>         subproblem.
!> 

GIVCOL

!>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
!>         Each pair of numbers indicates a pair of rows/columns
!>         involved in a Givens rotation.
!> 

LDGCOL

!>          LDGCOL is INTEGER
!>         The leading dimension of GIVCOL, must be at least N.
!> 

GIVNUM

!>          GIVNUM is REAL array, dimension ( LDGNUM, 2 )
!>         Each number indicates the C or S value used in the
!>         corresponding Givens rotation.
!> 

LDGNUM

!>          LDGNUM is INTEGER
!>         The leading dimension of arrays DIFR, POLES and
!>         GIVNUM, must be at least K.
!> 

POLES

!>          POLES is 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

!>          DIFL is 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

!>          DIFR is 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

!>          Z is REAL array, dimension ( K )
!>         Contain the components of the deflation-adjusted updating row
!>         vector.
!> 

K

!>          K is INTEGER
!>         Contains the dimension of the non-deflated matrix,
!>         This is the order of the related secular equation. 1 <= K <=N.
!> 

C

!>          C is 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

!>          S is 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

!>          RWORK is REAL array, dimension
!>         ( K*(1+NRHS) + 2*NRHS )
!> 

INFO

!>          INFO is INTEGER
!>          = 0:  successful exit.
!>          < 0:  if INFO = -i, the i-th argument had an illegal value.
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA