A Power-like Method for Computing the Dominant Eigenpairs of Large Scale Real Skew-Symmetric Matrices

TL;DR

This paper introduces a structure-preserving power-like method tailored for efficiently computing the dominant conjugate eigenpairs of large skew-symmetric matrices, with proven convergence and enhanced computational speed.

Contribution

It presents a novel power-like algorithm that preserves matrix structure, accelerates convergence, and includes a deflation technique for multiple eigenpairs, outperforming traditional methods.

Findings

Converges twice as fast for eigenvalues compared to eigenvectors.

Proven rigorous and quantitative convergence.

Effective in computing multiple eigenpairs with numerical experiments.

Abstract

The power method is a basic method for computing the dominant eigenpair of a matrix. In this paper, we propose a structure-preserving power-like method for computing the dominant conjugate pair of purely imaginary eigenvalues and the corresponding eigenvectors of a large skew-symmetric matrix S, which works on S and its transpose alternately and is performed in real arithmetic. We establish the rigorous and quantitative convergence of the proposed power-like method, and prove that the approximations to the dominant eigenvalues converge twice as fast as those to the associated eigenvectors. Moreover, we develop a deflation technique to compute several complex conjugate dominant eigenpairs of S. Numerical experiments show the effectiveness and efficiency of the new method.