# Disguised and new Quasi-Newton methods for nonlinear eigenvalue problems

## TL;DR

This paper develops new quasi-Newton algorithms for solving nonlinear eigenvalue problems, providing convergence analysis and unifying existing methods under a common framework.

## Contribution

It introduces novel quasi-Newton methods for NEPs, analyzes their convergence, and interprets existing algorithms as special cases of these methods.

## Key findings

- New quasi-Newton algorithms for NEPs are proposed.
- Convergence properties are established using Keldysh's theorem.
- Existing methods like residual inverse iteration are shown to be quasi-Newton methods.

## Abstract

In this paper we take a quasi-Newton approach to nonlinear eigenvalue problems (NEPs) of the type $M(\lambda)v=0$, where $M:\mathbb{C}\rightarrow\mathbb{C}^{n\times n}$ is a holomorphic function. We investigate which types of approximations of the Jacobian matrix lead to competitive algorithms, and provide convergence theory. The convergence analysis is based on theory for quasi-Newton methods and Keldysh's theorem for NEPs. We derive new algorithms and also show that several well-established methods for NEPs can be interpreted as quasi-Newton methods, and thereby provide insight to their convergence behavior. In particular, we establish quasi-Newton interpretations of Neumaier's residual inverse iteration and Ruhe's method of successive linear problems.