


DLARRB(3f)    LAPACK auxiliary routine (version 3.1)   DLARRB(3f)


NAME
       DLARRB  -  relatively  robust  representation(RRR) L D L^T, DLARRB does
       "limited" bisection to refine the eigenvalues of L D L^T,

SYNOPSIS
       SUBROUTINE DLARRB( N, D, LLD, IFIRST, ILAST, RTOL1, RTOL2,  OFFSET,  W,
                          WGAP, WERR, WORK, IWORK, PIVMIN, SPDIAM, TWIST, INFO
                          )

           INTEGER        IFIRST, ILAST, INFO, N, OFFSET, TWIST

           DOUBLE         PRECISION PIVMIN, RTOL1, RTOL2, SPDIAM

           INTEGER        IWORK( * )

           DOUBLE         PRECISION D( * ), LLD( * ), W( * ), WERR( * ), WGAP(
                          * ), WORK( * )

PURPOSE
       Given  the  relatively  robust representation(RRR) L D L^T, DLARRB does
       "limited" bisection to refine the eigenvalues of L D  L^T,  W(  IFIRST-
       OFFSET  )  through W( ILAST-OFFSET ), to more accuracy. Initial guesses
       for these eigenvalues are input in W, the corresponding estimate of the
       error  in  these  guesses  and  their  gaps are input in WERR and WGAP,
       respectively. During bisection, intervals
       [left, right] are maintained by  storing  their  mid-points  and  semi-
       widths in the arrays W and WERR respectively.


ARGUMENTS
       N       (input) INTEGER
               The order of the matrix.

       D       (input) DOUBLE PRECISION array, dimension (N)
               The N diagonal elements of the diagonal matrix D.

       LLD     (input) DOUBLE PRECISION array, dimension (N-1)
               The (N-1) elements L(i)*L(i)*D(i).

       IFIRST  (input) INTEGER
               The index of the first eigenvalue to be computed.

       ILAST   (input) INTEGER
               The index of the last eigenvalue to be computed.

       RTOL1   (input) DOUBLE PRECISION
               RTOL2    (input) DOUBLE PRECISION Tolerance for the convergence
               of the bisection intervals.  An interval [LEFT,RIGHT] has  con-
               verged         if         RIGHT-LEFT.LT.MAX(         RTOL1*GAP,
               RTOL2*MAX(|LEFT|,|RIGHT|) ) where GAP is the  (estimated)  dis-
               tance to the nearest eigenvalue.

       OFFSET  (input) INTEGER
               Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET



 LAPACK auxiliary routine November 2006                         1





DLARRB(3f)    LAPACK auxiliary routine (version 3.1)   DLARRB(3f)


               through ILAST-OFFSET elements of these arrays are to be used.

       W       (input/output) DOUBLE PRECISION array, dimension (N)
               On input, W( IFIRST-OFFSET )  through  W(  ILAST-OFFSET  )  are
               estimates  of  the eigenvalues of L D L^T indexed IFIRST throug
               ILAST.  On output, these estimates are refined.

       WGAP    (input/output) DOUBLE PRECISION array, dimension (N-1)
               On input, the (estimated) gaps between consecutive  eigenvalues
               of L D L^T, i.e., WGAP(I-OFFSET) is the gap between eigenvalues
               I and I+1. Note that if IFIRST.EQ.ILAST  then  WGAP(IFIRST-OFF-
               SET) must be set to ZERO.  On output, these gaps are refined.

       WERR    (input/output) DOUBLE PRECISION array, dimension (N)
               On  input,  WERR(  IFIRST-OFFSET ) through WERR( ILAST-OFFSET )
               are the errors in the estimates of the  corresponding  elements
               in W.  On output, these errors are refined.

       WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)
               Workspace.

       IWORK   (workspace) INTEGER array, dimension (2*N)
               Workspace.

       PIVMIN  (input) DOUBLE PRECISION
               The minimum pivot in the Sturm sequence.

       SPDIAM  (input) DOUBLE PRECISION
               The spectral diameter of the matrix.

       TWIST   (input) INTEGER
               The  twist index for the twisted factorization that is used for
               the negcount.  TWIST = N: Compute  negcount  from  L  D  L^T  -
               LAMBDA I = L+ D+ L+^T
               TWIST  =  1:  Compute  negcount from L D L^T - LAMBDA I = U- D-
               U-^T
               TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r)
               N(r)

       INFO    (output) INTEGER
               Error flag.

FURTHER DETAILS
       Based on contributions by
          Beresford Parlett, University of California, Berkeley, USA
          Jim Demmel, University of California, Berkeley, USA
          Inderjit Dhillon, University of Texas, Austin, USA
          Osni Marques, LBNL/NERSC, USA
          Christof Voemel, University of California, Berkeley, USA








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