dlalsd(3S)

NAME
     DLALSD - use the singular value decomposition of A to solve the least
     squares problem of finding X to minimize the Euclidean norm of each
     column of A*X-B, where A is N-by-N upper bidiagonal, and X and B are N-
     by-NRHS

SYNOPSIS
     SUBROUTINE DLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, RANK,
                        WORK, IWORK, INFO )

         CHARACTER      UPLO

         INTEGER        INFO, LDB, N, NRHS, RANK, SMLSIZ

         DOUBLE         PRECISION RCOND

         INTEGER        IWORK( * )

         DOUBLE         PRECISION B( LDB, * ), D( * ), E( * ), 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
     DLALSD uses the singular value decomposition of A to solve the least
     squares problem of finding X to minimize the Euclidean norm of each
     column of A*X-B, where A is N-by-N upper bidiagonal, and X and B are N-
     by-NRHS. The solution X overwrites B. The singular values of A smaller
     than RCOND times the largest singular value are treated as zero in
     solving the least squares problem; in this case a minimum norm solution
     is returned.  The actual singular values are returned in D in ascending
     order.

     This code makes very mild assumptions about floating point arithmetic. It
     will work on machines with a guard digit in add/subtract, or on those
     binary machines without guard digits which subtract like the Cray XMP,
     Cray YMP, Cray C 90, or Cray 2.  It could conceivably fail on hexadecimal
     or decimal machines without guard digits, but we know of none.

ARGUMENTS
     UPLO   (input) CHARACTER*1
            = 'U': D and E define an upper bidiagonal matrix.
            = 'L': D and E define a  lower bidiagonal matrix.

            SMLSIZ (input) INTEGER The maximum size of the subproblems at the
            bottom of the computation tree.

     N      (input) INTEGER
            The dimension of the  bidiagonal matrix.  N >= 0.

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

     D      (input/output) DOUBLE PRECISION array, dimension (N)
            On entry D contains the main diagonal of the bidiagonal matrix. On
            exit, if INFO = 0, D contains its singular values.

     E      (input) DOUBLE PRECISION array, dimension (N-1)
            Contains the super-diagonal entries of the bidiagonal matrix.  On
            exit, E has been destroyed.

     B      (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
            On input, B contains the right hand sides of the least squares
            problem. On output, B contains the solution X.

     LDB    (input) INTEGER
            The leading dimension of B in the calling subprogram.  LDB must be
            at least max(1,N).

     RCOND  (input) DOUBLE PRECISION
            The singular values of A less than or equal to RCOND times the
            largest singular value are treated as zero in solving the least
            squares problem. If RCOND is negative, machine precision is used
            instead.  For example, if diag(S)*X=B were the least squares
            problem, where diag(S) is a diagonal matrix of singular values,
            the solution would be X(i) = B(i) / S(i) if S(i) is greater than
            RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
            RCOND*max(S).

     RANK   (output) INTEGER
            The number of singular values of A greater than RCOND times the
            largest singular value.

     WORK   (workspace) DOUBLE PRECISION array, dimension at least
            (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2), where NLVL
            = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).

     IWORK  (workspace) INTEGER array, dimension at least
            (3*N*NLVL + 11*N)
     INFO   (output) INTEGER
            = 0:  successful exit.
            < 0:  if INFO = -i, the i-th argument had an illegal value.
            > 0:  The algorithm failed to compute an singular value while
            working on the submatrix lying in rows and columns INFO/(N+1)
            through MOD(INFO,N+1).

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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