# The infinite Lanczos method for symmetric nonlinear eigenvalue problems

## TL;DR

This paper introduces an infinite Lanczos method for symmetric nonlinear eigenvalue problems, utilizing an infinite-dimensional linearization and a specialized extraction process to improve computational efficiency and stability.

## Contribution

It develops a novel infinite-dimensional linearization and applies the indefinite Lanczos method with a tailored eigenpair extraction for symmetric nonlinear eigenproblems.

## Key findings

- Method achieves finite arithmetic under specific conditions
- Exploits problem structure for faster computation
- Addresses instability issues in eigenpair extraction

## Abstract

A new iterative method for solving large scale symmetric nonlinear eigenvalue problems is presented. We firstly derive an infinite dimensional symmetric linearization of the nonlinear eigenvalue problem, then we apply the indefinite Lanczos method to this specific linearization, resulting in a short-term recurrence. We show how, under specific assumption on the starting vector, this method can be carried out in finite arithmetic and how the exploitation of the problem structure leads to improvements in terms of computation time. The eigenpair approximations are extracted with the nonlinear Rayleigh-Ritz procedure combined with a specific choice of the projection space. We illustrate how this extraction technique resolves the instability issues that may occur due to the loss of orthogonality in many standard Lanczos-type methods.