发表机构
Universidad de Huelva（韦尔瓦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出对称酉核心追逐算法的矩阵推导，得到其无平方根变体，证明其向后稳定性，发现一般情况下严格结构化向后稳定性不可能，实验表明该算法可减少计算时间且精度不降低。
AI 中文摘要
对称矩阵和酉矩阵是最重要的结构化矩阵类别之一，可实现高效且稳定的特征值算法，典型例子包括QR算法的保结构变体及其无平方根对应算法。对于酉矩阵，这些算法历史上源于单位圆上正交多项式的递推关系；文献[3]中后来给出了基于核心追逐的矩阵推导，该算法族具有向后稳定性，但未识别出用于消除平方根的基础对称性，结构化向后稳定性的问题仍未解决。本文提出对称酉核心追逐算法的矩阵推导，并由此得到其无平方根变体；给出该算法向后稳定性的矩阵证明，证明一般情况下严格结构化向后稳定性不可能实现；开源Fortran代码和数值实验表明，利用对称性并消除平方根的组合可减少计算时间且不降低精度。
英文摘要
Symmetric and unitary matrices are among the most important classes of structured matrices admitting efficient and stable eigenvalue algorithms. Notable examples include structure-preserving variants of the QR algorithm and their square-root-free counterparts. For unitary matrices, these algorithms were historically derived from recurrence relations for orthogonal polynomials on the unit circle. A matrix-based derivation using core chasing was later given in [3]; this family of algorithms is backward stable, but the underlying symmetries needed to cast out the square roots were not identified, and the question of structured backward stability was left open. In this paper we present a matrix-based derivation of the symmetric unitary core-chasing algorithm and use it to obtain a square-root-free variant. We give a matrix-based proof of backward stability and show that strict structured backward stability is impossible in general. Open-source Fortran code and numerical experiments demonstrate that the combination of exploiting the symmetry and removing the square roots reduces computation time without degrading accuracy.
