# Compact representation of the full Broyden class of quasi-Newton updates

## TL;DR

This paper develops a compact representation for matrices in the Broyden class of quasi-Newton updates, enabling efficient computation of inverses, solutions to linear systems, and eigenvalues, thus facilitating sensitivity analysis.

## Contribution

It extends previous work by providing a compact representation for the full Broyden class, including rank-one and rank-two updates, with practical algorithms for inverse and linear system computations.

## Key findings

- Accurately represents Broyden class matrices using the compact form.
- Efficiently computes inverses and solves linear systems with these matrices.
- Enables eigenvalue, condition number, and sensitivity analysis for Broyden matrices.

## Abstract

In this paper, we present the compact representation for matrices belonging to the the Broyden class of quasi-Newton updates, where each update may be either rank-one or rank-two. This work extends previous results solely for the restricted Broyden class of rank-two updates. In this article, it is not assumed the same Broyden update is used each iteration; rather, different members of the Broyden class may be used each iteration. Numerical experiments suggest that a practical implementation of the compact representation is able to accurately represent matrices belonging to the Broyden class of updates. Furthermore, we demonstrate how to compute the compact representation for the inverse of these matrices, as well as a practical algorithm for solving linear systems with members of the Broyden class of updates. We demonstrate through numerical experiments that the proposed linear solver is able to efficiently solve linear systems with members of the Broyden class of matrices to high accuracy. As an immediate consequence of this work, it is now possible to efficiently compute the eigenvalues of any limited-memory member of the Broyden class of matrices, allowing for the computation of condition numbers and the ability perform sensitivity analysis.