larrb(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

Okay bestie, welcome to larrb(3) — straight from the Library Functions Manual — and it's literally the “step in stemr” page. Buckle up, we're getting into some eigenvalue-refinement lore. 🎀✨

💅 NAME

larrb — larrb: step in stemr

It's giving “a tiny but crucial step in stemr,” and that's the whole name. Slay. 💖

✨ 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

Lowkey there's no extra prose here — the whole page is the function documentation below. Stay with me, bestie. 🫶

🔥 FUNCTION DOCUMENTATION

🌟 subroutine dlarrb

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. It's the double-precision one, and yes she eats. 🔥

Purpose:

Given the relatively robust representation (RRR) L D L^T, DLARRB does 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. No cap. 💅

Parameters

  • N — INTEGER. The order of the matrix.
  • D — DOUBLE PRECISION array, dimension (N). The N diagonal elements of the diagonal matrix D.
  • LLD — DOUBLE PRECISION array, dimension (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 — DOUBLE PRECISION. That's the whole note in the original, bestie. 💅
  • RTOL2 — 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 — 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 — 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 — 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 — 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 — DOUBLE PRECISION array, dimension (2*N). Workspace.
  • IWORK — INTEGER array, dimension (2*N). Workspace.
  • PIVMIN — DOUBLE PRECISION. The minimum pivot in the Sturm sequence.
  • SPDIAM — DOUBLE PRECISION. 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 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 — 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

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. Same energy as dlarrb, but make it single precision. Iconic. 🌈

Purpose:

Given the relatively robust representation (RRR) L D L^T, SLARRB does 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. Periodt. 💅

Parameters

  • N — INTEGER. The order of the matrix.
  • D — REAL array, dimension (N). The N diagonal elements of the diagonal matrix D.
  • LLD — REAL array, dimension (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 — REAL. That's the whole note in the original again. She's mysterious. 🙈
  • RTOL2 — 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 — 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 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 — 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 — 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 — REAL array, dimension (2*N). Workspace.
  • IWORK — INTEGER array, dimension (2*N). Workspace.
  • PIVMIN — REAL. The minimum pivot in the Sturm sequence.
  • SPDIAM — REAL. 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 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 — 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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This page is larrb(3) from LAPACK, Version 3.12.0 — that's the footer tea, and there's no date listed, so I'm not making one up. Bye bestie, go refine those eigenvalues. 💖✨👑

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