?larrb2

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?larrb2

Provides limited bisection to locate eigenvalues for more accuracy.

Syntax

call slarrb2( n, d, lld, ifirst, ilast, rtol1, rtol2, offset, w, wgap, werr, work, iwork, pivmin, lgpvmn, lgspdm, twist, info )

call dlarrb2( n, d, lld, ifirst, ilast, rtol1, rtol2, offset, w, wgap, werr, work, iwork, pivmin, lgpvmn, lgspdm, twist, info )

Description

Given the relatively robust representation (RRR) LDL^T, ?larrb2 does "limited" bisection to refine the eigenvalues of LDL^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.

NOTE:

There are very few minor differences between larrb from LAPACK and this current subroutine ?larrb2. The most important reason for creating this nearly identical copy is profiling: in the ScaLAPACK MRRR algorithm, eigenvalue computation using ?larrb2 is used for refinement in the construction of the representation tree, as opposed to the initial computation of the eigenvalues for the root RRR which uses ?larrb. When profiling, this allows an easy quantification of refinement work vs. computing eigenvalues of the root.

Input Parameters

n

INTEGER

The order of the matrix.

d

REAL for slarrb2

DOUBLE PRECISION for dlarrb2

Array of size n.

The n diagonal elements of the diagonal matrix D.

lld

REAL for slarrb2

DOUBLE PRECISION for dlarrb2

Array of size n-1.

The (n-1) elements l_i*l_i*d(i).

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 slarrb2

DOUBLE PRECISION for dlarrb2

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

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

REAL for slarrb2

DOUBLE PRECISION for dlarrb2

Array of size n

On input, w( ifirst - offset ) through w( ilast- offset ) are estimates of the eigenvalues of LDL^T indexed ifirst through ilast.

wgap

REAL for slarrb2

DOUBLE PRECISION for dlarrb2

Array of size n-1.

On input, the (estimated) gaps between consecutive eigenvalues of LDL^T, 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.

werr

REAL for slarrb2

DOUBLE PRECISION for dlarrb2

Array of size 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 slarrb2

DOUBLE PRECISION for dlarrb2

(workspace) array of size 4*n.

Workspace.

iwork

(workspace) INTEGER array of size 2*n.

Workspace.

pivmin

REAL for slarrb2

DOUBLE PRECISION for dlarrb2

The minimum pivot in the Sturm sequence.

lgpvmn

REAL for slarrb2

DOUBLE PRECISION for dlarrb2

Logarithm of pivmin, precomputed.

lgspdm

REAL for slarrb2

DOUBLE PRECISION for dlarrb2

Logarithm of the spectral diameter, precomputed.

twist

INTEGER

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

twist = n: Compute negcount from LDL^T - λI = L₊D₊L₊^T

twist = 1: Compute negcount from LDL^T - λI = U_-D_-U_-^T

twist = r, 1 < r < n: Compute negcount from LDL^T - λI = N_r Δ_r N_r^T

OUTPUT Parameters

w

On output, the eigenvalue estimates in w are refined.

wgap

On output, the eigenvalue gaps in wgap are refined.

werr

On output, the errors in werr are refined.

info

INTEGER

Error flag.

Parent topic: ScaLAPACK Auxiliary Routines

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