dlals0(3S)

NAME
     DLALS0 - 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

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( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

     This man page is available only online.