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larrb

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine dlarrb (integer n, double precision, dimension( * ) d, doubleprecision, dimension( * ) lld, integer ifirst, integer ilast, doubleprecision rtol1, double precision rtol2, integer offset, doubleprecision, dimension( * ) w, double precision, dimension( * ) wgap,double precision, dimension( * ) werr, double precision, dimension( * )work, integer, dimension( * ) iwork, double precision pivmin, doubleprecision spdiam, integer twist, integer info)
subroutine slarrb (integer n, real, dimension( * ) d, real, dimension( * )lld, integer ifirst, integer ilast, real rtol1, real rtol2, integeroffset, real, dimension( * ) w, real, dimension( * ) wgap, real,dimension( * ) werr, real, dimension( * ) work, integer, dimension( * )iwork, real pivmin, real spdiam, integer twist, integer info)
Author

NAME

larrb - larrb: step in stemr

SYNOPSIS

Functions

subroutine dlarrb (n, d, lld, ifirst, ilast, rtol1, rtol2, offset, w, wgap, werr, work, iwork, pivmin, spdiam, twist, info)
DLARRB provides limited bisection to locate eigenvalues for more accuracy.
subroutine slarrb (n, d, lld, ifirst, ilast, rtol1, rtol2, offset, w, wgap, werr, work, iwork, pivmin, spdiam, twist, info)
SLARRB provides limited bisection to locate eigenvalues for more accuracy.

Detailed Description

Function Documentation

subroutine dlarrb (integer n, double precision, dimension( * ) d, doubleprecision, dimension( * ) lld, integer ifirst, integer ilast, doubleprecision rtol1, double precision rtol2, integer offset, doubleprecision, dimension( * ) w, double precision, dimension( * ) wgap,double precision, dimension( * ) werr, double precision, dimension( * )work, integer, dimension( * ) iwork, double precision pivmin, doubleprecision spdiam, integer twist, integer info)

DLARRB provides limited bisection to locate eigenvalues for more accuracy.

Purpose:

Given the relatively robust representation(RRR) L D LˆT, DLARRB
does ’limited’ bisection to refine the eigenvalues of L D LˆT,
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.

Parameters

N

N is INTEGER
The order of the matrix.

D

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

LLD

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

IFIRST

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

ILAST

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

RTOL1

RTOL1 is DOUBLE PRECISION

RTOL2

RTOL2 is DOUBLE PRECISION
Tolerance for the convergence of the bisection intervals.
An interval [LEFT,RIGHT] has converged if
RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
where GAP is the (estimated) distance to the nearest
eigenvalue.

OFFSET

OFFSET is INTEGER
Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET
through ILAST-OFFSET elements of these arrays are to be used.

W

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

WGAP

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

WERR

WERR is 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

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

IWORK

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

PIVMIN

PIVMIN is DOUBLE PRECISION
The minimum pivot in the Sturm sequence.

SPDIAM

SPDIAM is DOUBLE PRECISION
The spectral diameter of the matrix.

TWIST

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

INFO

INFO is INTEGER
Error flag.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

subroutine slarrb (integer n, real, dimension( * ) d, real, dimension( * )lld, integer ifirst, integer ilast, real rtol1, real rtol2, integeroffset, real, dimension( * ) w, real, dimension( * ) wgap, real,dimension( * ) werr, real, dimension( * ) work, integer, dimension( * )iwork, real pivmin, real spdiam, integer twist, integer info)

SLARRB provides limited bisection to locate eigenvalues for more accuracy.

Purpose:

Given the relatively robust representation(RRR) L D LˆT, SLARRB
does ’limited’ bisection to refine the eigenvalues of L D LˆT,
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.

Parameters

N

N is INTEGER
The order of the matrix.

D

D is REAL array, dimension (N)
The N diagonal elements of the diagonal matrix D.

LLD

LLD is REAL array, dimension (N-1)
The (N-1) elements L(i)*L(i)*D(i).

IFIRST

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

ILAST

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

RTOL1

RTOL1 is REAL

RTOL2

RTOL2 is REAL
Tolerance for the convergence of the bisection intervals.
An interval [LEFT,RIGHT] has converged if
RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
where GAP is the (estimated) distance to the nearest
eigenvalue.

OFFSET

OFFSET is INTEGER
Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET
through ILAST-OFFSET elements of these arrays are to be used.

W

W is REAL array, dimension (N)
On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
estimates of the eigenvalues of L D LˆT indexed IFIRST through
ILAST.
On output, these estimates are refined.

WGAP

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

WERR

WERR is REAL 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

WORK is REAL array, dimension (2*N)
Workspace.

IWORK

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

PIVMIN

PIVMIN is REAL
The minimum pivot in the Sturm sequence.

SPDIAM

SPDIAM is REAL
The spectral diameter of the matrix.

TWIST

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

INFO

INFO is INTEGER
Error flag.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Author

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