On efficiently computing the eigenvalues of limited-memory quasi-Newton matrices

TL;DR

This paper presents an efficient method for computing eigenvalues of limited-memory quasi-Newton matrices using QR updates, applicable to large-scale optimization and sensitivity analysis.

Contribution

It introduces a compact formula for Broyden convex class matrices and a QR-based update method that reduces computational cost and improves accuracy.

Findings

Method achieves high-accuracy eigenvalue computation.

Numerical experiments confirm efficiency and precision.

Applicable to large-scale optimization and condition number estimation.

Abstract

In this paper, we consider the problem of efficiently computing the eigenvalues of limited-memory quasi-Newton matrices that exhibit a compact formulation. In addition, we produce a compact formula for quasi-Newton matrices generated by any member of the Broyden convex class of updates. Our proposed method makes use of efficient updates to the QR factorization that substantially reduces the cost of computing the eigenvalues after the quasi-Newton matrix is updated. Numerical experiments suggest that the proposed method is able to compute eigenvalues to high accuracy. Applications for this work include modified quasi-Newton methods and trust-region methods for large-scale optimization, the efficient computation of condition numbers and singular values, and sensitivity analysis.