# A new shift strategy for the implicitly restarted generalized second-order Arnoldi method

## TL;DR

This paper introduces a novel shift strategy for the implicitly restarted GSOAR method that effectively utilizes all shift candidates while preserving structure, significantly improving algorithm efficiency.

## Contribution

A new shift strategy that uses all shift candidates in the implicitly restarted GSOAR method without destroying its structure, enhancing computational efficiency.

## Key findings

- The new method preserves GSOAR structure with all shift candidates.
- Numerical experiments demonstrate improved efficiency of the restart process.
- The approach outperforms existing strategies in computational tests.

## Abstract

In this paper, a new shift strategy for the implicitly restarted generalized second-order Arnoldi (GSOAR) method is proposed. In implicitly restarted processes, we can get a $k$-step GSOAR decomposition from a $m$-step GSOAR decomposition by performing $p = m-k$ implicit shifted QR iterations. The problem of the implicitly restarted GSOAR is the mismatch between the number of shifts and the dimension of the subspace. There are $2p$ shifts for $p$ QR iterations. We use the shifts to filter out the unwanted information in the current subspace; when more shifts are used, one obtains a better updated subspace. But, if we use more than $p$ shifts, the structure of the GSOAR decomposition will be destroyed. We propose a novel method which can use all $2p$ candidates and preserve the special structure. The new method vastly enhances the overall efficiency of the algorithm. Numerical experiments illustrate the efficiency of every restart process.