A thick-restart Lanczos type algorithm is proposed for Hermitian $J$-symmetric matrices. Since Hermitian $J$-symmetric matrices possess doubly degenerate spectra or doubly multiple eigenvalues with a simple relation between the degenerate eigenvectors, we can improve the convergence of the Lanczos algorithm by restricting the search space of the Krylov subspace to that spanned by one of each pair of the degenerate eigenvector pairs. We show that the Lanczos iteration is compatible with the $J$-symmetry, so that the subspace can be split into two subspaces that are orthogonal to each other. The proposed algorithm searches for eigenvectors in one of the two subspaces without the multiplicity. The other eigenvectors paired to them can be easily reconstructed with the simple relation from the $J$-symmetry. We test our algorithm on randomly generated small dense matrices and a sparse large matrix originating from a quantum field theory.

中文翻译：

---

Hermitian J-对称特征值问题的一种厚重启动Lanczos型方法

针对Hermitian$J$对称矩阵提出了一种厚重启动Lanczos型算法。由于 Hermitian $J$-对称矩阵具有双重退化谱或双重多重特征值，并且退化特征向量之间具有简单的关系，因此我们可以通过将 Krylov 子空间的搜索空间限制为每一个所跨越的搜索空间来提高 Lanczos 算法的收敛性一对退化的特征向量对。我们证明了 Lanczos 迭代与 $J$-对称兼容，因此子空间可以拆分为两个相互正交的子空间。所提出的算法在没有多重性的两个子空间之一中搜索特征向量。与它们配对的其他特征向量可以通过 $J$ 对称性的简单关系轻松重建。