# The inexact residual iteration method for quadratic eigenvalue problem and the analysis of convergence

## TL;DR

This paper develops and analyzes an inexact residual iteration method for quadratic eigenvalue problems, incorporating inner-outer iteration strategies to improve efficiency for large-scale problems, with proven convergence criteria and numerical validation.

## Contribution

It introduces an inexact residual iteration approach with convergence analysis and criteria for large-scale quadratic eigenvalue problems, enhancing computational efficiency.

## Key findings

- Convergence criteria established for the residual iteration method.
- Inner-outer iteration relationship analyzed and quantified.
- Numerical experiments confirm the method's effectiveness.

## Abstract

In this paper, we first establish the convergence criteria of the residual iteration method for solving quadratic eigenvalue problem- s. We analyze the impact of shift point and the subspace expansion on the convergence of this method. In the process of expanding subspace, this method needs to solve a linear system at every step. For large scale problems in which the equations cannot be solved directly, we propose an inner and outer iteration version of the residual iteration method. The new method uses the iterative method to solve the equations and uses the approximate solution to expand the subspace. We analyze the relationship between inner and outer iterations and provide a quantita- tive criterion for the inner iteration which can ensure the convergence of the outer iteration. Finally, our numerical experiments provide proof of our analysis and demonstrate the effectiveness of the inexact residual iteration method.