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<p>In this paper, we propose and develop two variants of the power method that are performed in <i>real</i> arithmetic for computing the complex conjugate dominant eigenpairs of a large real skew-symmetric matrix. The first variant is from Wilkinson’s 1965 classic book “The Algebraic Eigenvalue Problem", which formulates the near linear dependence of the consecutive three iterates obtained by the power method as a least squares problem, but does not give a precise definition of near linear dependence and any relationship between the near linear dependence and stopping criteria. Instead of using the least squares to determine the near linear dependence of the consecutive three iterates, based on the singular value decomposition (SVD) of the small matrix consisting of three iterates, we propose a new variant. These two variants only use real arithmetic to compute the complex conjugate dominant eigenpairs of a general real matrix and much simplify for the skew-symmetric <i>S</i>. For the proposed power-like method and Wilkinson’s original method, we derive explicit relationships between the stopping criteria and the near linear dependence, based on which we design general-purpose stopping criteria. We also establish the rigorous and quantitative convergence of the proposed methods. Numerical experiments confirm the effectiveness of the two power-like methods.</p>

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Least squares and SVD based power-like methods for computing the dominant eigenpairs of large scale real skew-symmetric matrices

  • 期刊论文

  • 发表日期：

    2025年12月18日

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  • Qingqing Zheng

摘要

In this paper, we propose and develop two variants of the power method that are performed in real arithmetic for computing the complex conjugate dominant eigenpairs of a large real skew-symmetric matrix. The first variant is from Wilkinson’s 1965 classic book “The Algebraic Eigenvalue Problem", which formulates the near linear dependence of the consecutive three iterates obtained by the power method as a least squares problem, but does not give a precise definition of near linear dependence and any relationship between the near linear dependence and stopping criteria. Instead of using the least squares to determine the near linear dependence of the consecutive three iterates, based on the singular value decomposition (SVD) of the small matrix consisting of three iterates, we propose a new variant. These two variants only use real arithmetic to compute the complex conjugate dominant eigenpairs of a general real matrix and much simplify for the skew-symmetric S. For the proposed power-like method and Wilkinson’s original method, we derive explicit relationships between the stopping criteria and the near linear dependence, based on which we design general-purpose stopping criteria. We also establish the rigorous and quantitative convergence of the proposed methods. Numerical experiments confirm the effectiveness of the two power-like methods.