# New numerical algorithm for deflation of infinite and zero eigenvalues and full solution of quadratic eigenvalue problems

## TL;DR

This paper introduces an improved numerical algorithm for accurately computing all eigenvalues and eigenvectors of quadratic matrix pencils, especially handling zero and infinite eigenvalues more robustly and efficiently.

## Contribution

It enhances the existing quadeig algorithm by improving stability, robustness, and the ability to deflate additional infinite eigenvalues using a new canonical form approach.

## Key findings

- Enhanced backward stability and numerical robustness.
- Ability to deflate more infinite eigenvalues.
- Superior performance demonstrated through empirical testing.

## Abstract

This paper presents a new method for computing all eigenvalues and eigenvectors of quadratic matrix pencil. It is an upgrade of the quadeig algorithm by Hammarling, Munro and Tisseur, which attempts to reveal and remove by deflation certain number of zero and infinite eigenvalues before QZ iterations. Proposed modifications of the quadeig framework are designed to enhance backward stability and to make the process of deflating infinite and zero eigenvalues more numerically robust. In particular, careful preprocessing allows scaling invariant/component-wise backward error and thus better condition number. Further, using an upper triangular version of the Kronecker canonical form enables deflating additional infinite eigenvalues, in addition to those inferred from the rank of leading coefficient matrix. Theoretical analysis and empirical evidence from thorough testing of the software implementation confirm superior numerical performances of the proposed method.