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

TL;DR

Problem: computing the dominant complex-conjugate eigenpairs of large real skew-symmetric matrices in real arithmetic. Approach: a structure-preserving skew-symmetric power (SSP) method that alternates applying $S$ and $S^T$ to produce real and imaginary parts of the dominant eigenvector, with rigorous convergence results and a deflation framework to obtain multiple pairs. Key contributions include explicit convergence bounds showing eigenvalue residuals converge twice as fast as eigenvectors, a practical stopping criterion, and a deflation-based algorithm to compute several complex-conjugate dominant eigenpairs while preserving skew-symmetric structure. Numerical experiments on large-scale problems validate effectiveness and scalability, confirming theoretical rates and demonstrating the method's practicality for applications requiring skew-symmetric structure preservation.

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.

Figures (4)

• Figure 1: (a): convergence curves of the relative residual norms of \ref{['alg-MpowerSTS']} for $l=32$, $s=5$; (b): the aveCPU for $s=1,2,\ldots,5$.
• Figure 2: Curves of $\log(|\tan\angle(x_{2k-1},x_{o})|)$ (circle ), $\log(|\tan\angle(x_{2k},x_{e})|)$ (triangle ), $\log(|\rho_{k}-\sigma_{1}|)$ (plus ), $\log((\dfrac{\sigma_{2}}{\sigma_{1}})^{2k}|\tan\angle(q_{0},x_{e})|)$ (dashed line ) and $\log((\dfrac{\sigma_{2}}{\sigma_{1}})^{2k-1}|\tan\angle(q_{0},x_{e})|)$ (solid line ) for the matrix $S$ in Example \ref{['exa1']} with $l=4$.
• Figure 3: (a) and (b): convergence curves of the relative residual norms of \ref{['alg-MpowerSTS']} for plsk1919 and plskz362, respectively.
• Figure 4: The aveCPU for each complex conjugate dominant eigenpair for $s$ from $1$ to $5$.

Theorems & Definitions (17)

• Proposition 2.1
• Theorem 2.2
• Proof 1
• Remark 2.3
• Remark 2.4
• Lemma 3.1
• Proof 2
• Lemma 3.2
• Proof 3
• Remark 3.3