2D Eigenvalue Problem II: Rayleigh Quotient Iteration and Applications

TL;DR

This paper introduces a new Rayleigh quotient iteration algorithm for the 2D eigenvalue problem, significantly improving computational speed for large-scale eigenvalue optimization tasks.

Contribution

It develops the 2DRQI algorithm, which is faster than existing methods for solving the 2D eigenvalue problem in large-scale applications.

Findings

2DRQI is 2 to 8 times faster than existing algorithms.

The method effectively solves large-scale eigenvalue optimization problems.

It enhances computational efficiency in eigenvalue-related stability analysis.

Abstract

In Part I of this paper, we introduced a 2D eigenvalue problem (2DEVP) and presented theoretical results of the 2DEVP and its intrinsic connetion with the eigenvalue optimizations. In this part, we devise a Rayleigh quotient iteration (RQI)-like algorithm, 2DRQI in short, for computing a 2D-eigentriplet of the 2DEVP. The 2DRQI performs $2\times$ to $8\times$ faster than the existing algorithms for large scale eigenvalue optimizations arising from the minmax of Rayleigh quotients and the distance to instability of a stable matrix.