Complexity of Path-Following Methods for the Eigenvalue Problem

TL;DR

This paper introduces a framework to analyze the complexity of path-following methods for eigenvalue problems, relating condition numbers to problem difficulty and providing bounds on algorithmic complexity.

Contribution

It develops a unitarily invariant framework, defines a new condition number, and proves a complexity bound based on the path length in the condition metric.

Findings

Bound on path-following complexity in terms of path length

Relation between condition number and distance to ill-posedness

A version of Smale's γ-Theorem for eigenvalue problems

Abstract

A unitarily invariant projective framework is introduced to analyze the complexity of path-following methods for the eigenvalue problem. A condition number, and its relation to the distance to ill-posedness, is given. A Newton map appropriate for this context is defined, and a version of Smale's $\gamma$-Theorem is proven. The main result of this paper bounds the complexity of path-following methods in terms of the length of the path in the condition metric.