# A multishift, multipole rational QZ method with aggressive early deflation

## TL;DR

This paper introduces an advanced multishift, multipole rational QZ method with aggressive early deflation, significantly improving efficiency and accuracy in solving dense generalized eigenvalue problems.

## Contribution

It extends the rational QZ method by incorporating higher multiplicity shifts and poles, enabling real arithmetic and enhanced deflation techniques.

## Key findings

- Competitive speed compared to state-of-the-art routines
- High accuracy in eigenvalue computations
- Effective handling of dense generalized eigenvalue problems

## Abstract

The rational QZ method generalizes the QZ method by implicitly supporting rational subspace iteration. In this paper we extend the rational QZ method by introducing shifts and poles of higher multiplicity in the Hessenberg pencil, which is a pencil consisting of two Hessenberg matrices. The result is a multishift, multipole iteration on block Hessenberg pencils which allows one to stick to real arithmetic for a real input pencil. In combination with optimally packed shifts and aggressive early deflation as an advanced deflation technique we obtain an efficient method for the dense generalized eigenvalue problem. In the numerical experiments we compare the results with state-of-the-art routines for the generalized eigenvalue problem and show that we are competitive in terms of speed and accuracy.

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Source: https://tomesphere.com/paper/1902.10954