On solving large-scale limited-memory quasi-Newton equations

TL;DR

This paper introduces a new practical method for efficiently solving large-scale linear systems involving limited-memory quasi-Newton matrices, improving speed and accuracy over existing algorithms.

Contribution

It presents a novel approach based on a compact representation for the inverse of limited-memory matrices, enhancing computational efficiency and enabling easy condition number estimation.

Findings

Method compares favorably in speed and accuracy

Efficient computation of system matrix condition number

Competitive with update-specific algorithms

Abstract

We consider the problem of solving linear systems of equations arising with limited-memory members of the restricted Broyden class of updates and the symmetric rank-one (SR1) update. In this paper, we propose a new approach based on a practical implementation of the compact representation for the inverse of these limited-memory matrices. Numerical results suggest that the proposed method compares favorably in speed and accuracy to other algorithms and is competitive with several update-specific methods available to only a few members of the Broyden class of updates. Using the proposed approach has an additional benefit: The condition number of the system matrix can be computed efficiently.