# ?larrb

Provides limited bisection to locate eigenvalues for more accuracy.

## Syntax

call slarrb( n, d, lld, ifirst, ilast, rtol1, rtol2, offset, w, wgap, werr, work, iwork, pivmin, spdiam, twist, info )

call dlarrb( n, d, lld, ifirst, ilast, rtol1, rtol2, offset, w, wgap, werr, work, iwork, pivmin, spdiam, twist, info )

## Description

Given the relatively robust representation (RRR) L*D*LT, the routine does “limited” bisection to refine the eigenvalues of L*D*LT, 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.

## Input Parameters

n

  INTEGER. The order of the matrix.

d

  REAL for slarrb

  DOUBLE PRECISION for dlarrb

  Array, DIMENSION (n). The n diagonal elements of the diagonal matrix D.

lld

  REAL for slarrb

  DOUBLE PRECISION for dlarrb

  Array, DIMENSION (n-1).

  The n-1 elements `Li*Li*Di`.

ifirst

  INTEGER. The index of the first eigenvalue to be computed.

ilast

  INTEGER. The index of the last eigenvalue to be computed.

rtol1, rtol2

  REAL for slarrb

  DOUBLE PRECISION for dlarrb

  Tolerance for the convergence of the bisection intervals. An interval [left, right] has converged if `RIGHT-LEFT.LT.MAX( rtol1*gap, rtol2*max(|left|,|right|) )`, where gap is the (estimated) distance to the nearest eigenvalue.

offset

  INTEGER. Offset for the arrays w, wgap and werr, that is, the ifirst-offset through ilast-offset elements of these arrays are to be used.

w

  REAL for slarrb

  DOUBLE PRECISION for dlarrb

  Array, DIMENSION (n). On input, w( ifirst-offset ) through w( ilast-offset ) are estimates of the eigenvalues of L*D*LT indexed ifirst through ilast.

wgap

  REAL for slarrb

  DOUBLE PRECISION for dlarrb

  Array, DIMENSION (n-1). The estimated gaps between consecutive eigenvalues of L*D*LT, that is, wgap(i-offset) is the gap between eigenvalues i and i+1. Note that if IFIRST.EQ.ILAST then wgap(ifirst-offset) must be set to 0.

werr

  REAL for slarrb

  DOUBLE PRECISION for dlarrb

  Array, DIMENSION (n). On input, werr(ifirst-offset) through werr(ilast-offset) are the errors in the estimates of the corresponding elements in w.

work

  REAL for slarrb

  DOUBLE PRECISION for dlarrb

  Workspace array, DIMENSION (2*n).

pivmin

  REAL for slarrb

  DOUBLE PRECISION for dlarrb

  The minimum pivot in the Sturm sequence.

spdiam

  REAL for slarrb

  DOUBLE PRECISION for dlarrb

  The spectral diameter of the matrix.

twist

  INTEGER. The twist index for the twisted factorization that is used for the negcount.

  twist = n: Compute negcount from `L*D*LT - lambda*i = L+* D+ *L+T`

  twist = n: Compute negcount from `L*D*LT - lambda*i = U-*D-*U-T`

  twist = n: Compute negcount from `L*D*LT - lambda*i = Nr*D r*Nr`

iwork

  INTEGER.

  Workspace array, DIMENSION (2*n).

## Output Parameters

w

  On output, the estimates of the eigenvalues are“refined”.

wgap

  On output, the gaps are refined.

werr

  On output, “refined” errors in the estimates of w.

info

  INTEGER.

  Error flag.

Parent topic: Auxiliary Routines