slasd1(3S)

NAME
     SLASD1 - compute the SVD of an upper bidiagonal N-by-M matrix B,

SYNOPSIS
     SUBROUTINE SLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT, IDXQ,
                        IWORK, WORK, INFO )

         INTEGER        INFO, LDU, LDVT, NL, NR, SQRE

         REAL           ALPHA, BETA

         INTEGER        IDXQ( * ), IWORK( * )

         REAL           D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )

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
     SLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B, where N =
     NL + NR + 1 and M = N + SQRE. SLASD1 is called from SLASD0.

     A related subroutine SLASD7 handles the case in which the singular values
     (and the singular vectors in factored form) are desired.

     SLASD1 computes the SVD as follows:

                   ( D1(in)  0    0     0 )
       B = U(in) * (   Z1'   a   Z2'    b ) * VT(in)
                   (   0     0   D2(in) 0 )

         = U(out) * ( D(out) 0) * VT(out)

     where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M with
     ALPHA and BETA in the NL+1 and NL+2 th entries and zeros elsewhere; and
     the entry b is empty if SQRE = 0.

     The left singular vectors of the original matrix are stored in U, and the
     transpose of the right singular vectors are stored in VT, and the
     singular values are in D.  The algorithm consists of three stages:

        The first stage consists of deflating the size of the problem
        when there are multiple singular values or when there are zeros in
        the Z vector.  For each such occurence the dimension of the
        secular equation problem is reduced by one.  This stage is
        performed by the routine SLASD2.

        The second stage consists of calculating the updated
        singular values. This is done by finding the square roots of the
        roots of the secular equation via the routine SLASD4 (as called
        by SLASD3). This routine also calculates the singular vectors of
        the current problem.

        The final stage consists of computing the updated singular vectors
        directly using the updated singular values.  The singular vectors
        for the current problem are multiplied with the singular vectors
        from the overall problem.


ARGUMENTS
     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.

     D      (input/output) REAL array,
            dimension (N = NL+NR+1).  On entry D(1:NL,1:NL) contains the
            singular values of the
            upper block; and D(NL+2:N) contains the singular values of
            the lower block. On exit D(1:N) contains the singular values of
            the modified matrix.

     ALPHA  (input) REAL
            Contains the diagonal element associated with the added row.

     BETA   (input) REAL
            Contains the off-diagonal element associated with the added row.

     U      (input/output) REAL array, dimension(LDU,N)
            On entry U(1:NL, 1:NL) contains the left singular vectors of
            the upper block; U(NL+2:N, NL+2:N) contains the left singular
            vectors of the lower block. On exit U contains the left singular
            vectors of the bidiagonal matrix.

     LDU    (input) INTEGER
            The leading dimension of the array U.  LDU >= max( 1, N ).

     VT     (input/output) REAL array, dimension(LDVT,M)
            where M = N + SQRE.  On entry VT(1:NL+1, 1:NL+1)' contains the
            right singular
            vectors of the upper block; VT(NL+2:M, NL+2:M)' contains the right
            singular vectors of the lower block. On exit VT' contains the
            right singular vectors of the bidiagonal matrix.

     LDVT   (input) INTEGER
            The leading dimension of the array VT.  LDVT >= max( 1, M ).

     IDXQ  (output) INTEGER array, dimension(N)
           This contains the permutation which will reintegrate the subproblem
           just solved back into sorted order, i.e.  D( IDXQ( I = 1, N ) )
           will be in ascending order.

     IWORK  (workspace) INTEGER array, dimension( 4 * N )

     WORK   (workspace) REAL array, dimension( 3*M**2 + 2*M )

     INFO   (output) INTEGER
            = 0:  successful exit.
            < 0:  if INFO = -i, the i-th argument had an illegal value.
            > 0:  if INFO = 1, an singular value did not converge

FURTHER DETAILS
     Based on contributions by
        Ming Gu and Huan Ren, Computer Science Division, University of
        California at Berkeley, USA


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

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