A stable, polynomial-time algorithm for the eigenpair problem

TL;DR

This paper presents a numerically stable, polynomial-time algorithm for computing eigenpairs of complex matrices, addressing a long-standing open problem in numerical linear algebra.

Contribution

It introduces a theoretically efficient algorithm for eigenpair computation that is stable and accurate, solving a major open problem.

Findings

Algorithms are numerically stable and strongly accurate.

They are theoretically efficient with polynomial-time complexity.

Practically, they may not outperform existing methods.

Abstract

We describe algorithms for computing eigenpairs (eigenvalue-eigenvector pairs) of a complex $n\times n$ matrix $A$. These algorithms are numerically stable, strongly accurate, and theoretically efficient (i.e., polynomial-time). We do not believe they outperform in practice the algorithms currently used for this computational problem. The merit of our paper is to give a positive answer to a long-standing open problem in numerical linear algebra.