一种计算大规模实反对称矩阵主导特征对的类幂法
A Power-like Method for Computing the Dominant Eigenpairs of Large Scale Real Skew-Symmetric Matrices
AI总结：
针对大规模实反对称矩阵，提出交替作用于矩阵及其转置的保结构类幂法与收缩技术，可在实数运算中高效计算多个纯虚主导共轭特征对，并证明特征值近似比特征向量近似快两倍收敛。
摘要：
幂法是计算矩阵主导特征对的一种基本方法。本文提出一种保结构的类幂法，用于计算大型反对称矩阵 S 的主导纯虚特征值共轭对及相应特征向量；该方法交替作用于 S 及其转置，并在实数运算中执行。我们建立了所提类幂法严格且定量的收敛性，证明主导特征值近似值的收敛速度是相关特征向量近似值的两倍。此外，我们发展了一种收缩技术，用于计算 S 的若干复共轭主导特征对。数值实验表明了新方法的有效性和效率。
英文摘要：
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.
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