Scalable computation of Jordan chains

TL;DR

This paper introduces a scalable inverse-iteration algorithm for computing Jordan chains of nearly defective matrices, suitable for large-scale problems, with proven accuracy and potential for higher-order convergence.

Contribution

The paper presents a novel inverse-iteration based algorithm for Jordan chain computation that is efficient for large matrices and does not rely on SVD, unlike previous methods.

Findings

Algorithm achieves $O( ext{ extsterling})$ error in eigenvector and Jordan vector

Demonstrated on a large electromagnetism problem with $212^2 imes 212^2$ matrix

Extension allows higher order convergence if matrix derivative is known

Abstract

We present an algorithm to compute the Jordan chain of a nearly defective matrix with a $2\times2$ Jordan block. The algorithm is based on an inverse-iteration procedure and only needs information about the invariant subspace corresponding to the Jordan chain, making it suitable for use with large matrices arising in applications, in contrast with existing algorithms which rely on an SVD. The algorithm produces the eigenvector and Jordan vector with $O(\varepsilon)$ error, with $\varepsilon$ being the distance of the given matrix to an exactly defective matrix. As an example, we demonstrate the use of this algorithm in a problem arising from electromagnetism, in which the matrix has size $212^2\times 212^2$. An extension of this algorithm is also presented which can achieve higher order convergence [$O(\varepsilon^2)$] when the matrix derivative is known.