# larrb - Man Page

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

**DLARRB** provides limited bisection to locate eigenvalues for more accuracy.

**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.

**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 L^T 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 L^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. 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 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*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

Definition at line **193** of file **dlarrb.f**.

### subroutine slarrb (integer n, real, dimension( * ) d, real, dimension( * ) lld, integer ifirst, integer ilast, real rtol1, real rtol2, integer offset, 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 L^T, SLARRB 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.

**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 L^T 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 L^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. 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 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*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

Definition at line **193** of file **slarrb.f**.

## Author

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