CLALS0(3)      LAPACK routine of NEC Numeric Library Collection      CLALS0(3)



NAME
       CLALS0

SYNOPSIS
       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)



PURPOSE
            CLALS0 applies back the multiplying factors of either the left or the
            right singular vector matrix of a bidiagonal 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)
                     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        (input)
                     NL is INTEGER
                    The row dimension of the upper block. NL >= 1.

           NR        (input)
                     NR is INTEGER
                    The row dimension of the lower block. NR >= 1.

           SQRE      (input)
                     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      (input)
                     NRHS is INTEGER
                    The number of columns of B and BX. NRHS must be at least 1.

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

           BX        (output)
                     BX is COMPLEX array, dimension ( LDBX, NRHS )

           LDBX      (input)
                     LDBX is INTEGER
                    The leading dimension of BX.

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

           GIVPTR    (input)
                     GIVPTR is INTEGER
                    The number of Givens rotations which took place in this
                    subproblem.

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

           LDGCOL    (input)
                     LDGCOL is INTEGER
                    The leading dimension of GIVCOL, must be at least N.

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

           LDGNUM    (input)
                     LDGNUM is INTEGER
                    The leading dimension of arrays DIFR, POLES and
                    GIVNUM, must be at least K.

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

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

           C         (input)
                     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         (input)
                     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     (output)
                     RWORK is REAL array, dimension
                    ( K*(1+NRHS) + 2*NRHS )

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



LAPACK routine                  31 October 2017                      CLALS0(3)