通过极分解将斜对称特征值问题的规模减半
Halving the size of skew-symmetric eigenvalue problems via the polar decomposition
浏览论文内容
中文总结 AI 辅助
本文提出一种新算法，利用斜对称正交极因子将稠密实斜对称矩阵的特征值问题规模减半，可通过LAPACK高效求解，稳定性与运行时间具竞争力，还可推广到正交特征值问题的规模约化。
中文摘要 AI 辅助
本文提出一种用于计算稠密实斜对称矩阵$A$的特征值与特征向量的新算法，其核心是计算$A$的一个同时兼具斜对称性与正交性的极因子，该极因子被用于将原问题转化为规模减半的Hermitian特征值问题，可通过LAPACK等标准软件精确高效求解。数值实验验证了该方法的稳定性，表明其运行时间与现有斜对称特征值问题方法相比具有竞争力。最后，本文证明相同原理可用于将正交特征值问题约化为规模减半的酉特征值问题。
英文摘要
This paper introduces a novel algorithm for computing eigenvalues and eigenvectors of a dense real skew-symmetric matrix $A$. Its main ingredient is the computation of a polar factor of $A$ that is both skew-symmetric and orthogonal. This polar factor is then used to transform the original problem into a Hermitian eigenvalue problem of half the size, which can be solved accurately and efficiently with standard software such as LAPACK. Numerical experiments demonstrate the stability of the method and show that its running time is competitive with existing approaches for skew-symmetric eigenvalue problems. Finally, we show that the same principle can be used to reduce an orthogonal eigenvalue problem to a unitary eigenvalue problem of half the size.
