clalsa(3S)

NAME
     CLALSA - i an itermediate step in solving the least squares problem by
     computing the SVD of the coefficient matrix in compact form (The singular
     vectors are computed as products of simple orthorgonal matrices.)

SYNOPSIS
     SUBROUTINE CLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U, LDU, VT,
                        K, DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, PERM,
                        GIVNUM, C, S, RWORK, IWORK, INFO )

         INTEGER        ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS, SMLSIZ

         INTEGER        GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), K( * ),
                        PERM( LDGCOL, * )

         REAL           C( * ), DIFL( LDU, * ), DIFR( LDU, * ), GIVNUM( LDU, *
                        ), POLES( LDU, * ), RWORK( * ), S( * ), U( LDU, * ),
                        VT( LDU, * ), Z( LDU, * )

         COMPLEX        B( LDB, * ), BX( LDBX, * )

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
     CLALSA is an itermediate step in solving the least squares problem by
     computing the SVD of the coefficient matrix in compact form (The singular
     vectors are computed as products of simple orthorgonal matrices.). If
     ICOMPQ = 0, CLALSA applies the inverse of the left singular vector matrix
     of an upper bidiagonal matrix to the right hand side; and if ICOMPQ = 1,
     CLALSA applies the right singular vector matrix to the right hand side.
     The singular vector matrices were generated in compact form by CLALSA.


ARGUMENTS
     ICOMPQ (input) INTEGER Specifies whether the left or the right singular
     vector matrix is involved.  = 0: Left singular vector matrix
     = 1: Right singular vector matrix

     SMLSIZ (input) INTEGER The maximum size of the subproblems at the bottom
     of the computation tree.
     N      (input) INTEGER
            The row and column dimensions of the upper bidiagonal matrix.

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

     B      (input) 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 in the calling subprogram.  LDB must be
            at least max(1,MAX( M, N ) ).

     BX     (output) COMPLEX array, dimension ( LDBX, NRHS )
            On exit, the result of applying the left or right singular vector
            matrix to B.

     LDBX   (input) INTEGER
            The leading dimension of BX.

     U      (input) REAL array, dimension ( LDU, SMLSIZ ).
            On entry, U contains the left singular vector matrices of all
            subproblems at the bottom level.

     LDU    (input) INTEGER, LDU = > N.
            The leading dimension of arrays U, VT, DIFL, DIFR, POLES, GIVNUM,
            and Z.

     VT     (input) REAL array, dimension ( LDU, SMLSIZ+1 ).
            On entry, VT' contains the right singular vector matrices of all
            subproblems at the bottom level.

     K      (input) INTEGER array, dimension ( N ).

     DIFL   (input) REAL array, dimension ( LDU, NLVL ).
            where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.

     DIFR   (input) REAL array, dimension ( LDU, 2 * NLVL ).
            On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record distances
            between singular values on the I-th level and singular values on
            the (I -1)-th level, and DIFR(*, 2 * I) record the normalizing
            factors of the right singular vectors matrices of subproblems on
            I-th level.

     Z      (input) REAL array, dimension ( LDU, NLVL ).
            On entry, Z(1, I) contains the components of the deflation-
            adjusted updating row vector for subproblems on the I-th level.

     POLES  (input) REAL array, dimension ( LDU, 2 * NLVL ).
            On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
            singular values involved in the secular equations on the I-th
            level.

            GIVPTR (input) INTEGER array, dimension ( N ).  On entry, GIVPTR(
            I ) records the number of Givens rotations performed on the I-th
            problem on the computation tree.

            GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 * NLVL ).  On
            entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the
            locations of Givens rotations performed on the I-th level on the
            computation tree.

            LDGCOL (input) INTEGER, LDGCOL = > N.  The leading dimension of
            arrays GIVCOL and PERM.

     PERM   (input) INTEGER array, dimension ( LDGCOL, NLVL ).
            On entry, PERM(*, I) records permutations done on the I-th level
            of the computation tree.

            GIVNUM (input) REAL array, dimension ( LDU, 2 * NLVL ).  On entry,
            GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S- values of Givens
            rotations performed on the I-th level on the computation tree.

     C      (input) REAL array, dimension ( N ).
            On entry, if the I-th subproblem is not square, C( I ) contains
            the C-value of a Givens rotation related to the right null space
            of the I-th subproblem.

     S      (input) REAL array, dimension ( N ).
            On entry, if the I-th subproblem is not square, S( I ) contains
            the S-value of a Givens rotation related to the right null space
            of the I-th subproblem.

     RWORK  (workspace) REAL array, dimension at least
            max ( N, (SMLSZ+1)*NRHS*3 ).

     IWORK  (workspace) INTEGER array.
            The dimension must be at least 3 * N

     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)

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