隐式重启广义二阶Arnoldi方法的一种新移位策略
A new shift strategy for the implicitly restarted generalized second-order Arnoldi method
AI总结：
本文提出了一种改进的隐式重启广义二阶Arnoldi方法的移位策略，通过优化移位数量与子空间维度的匹配，提升算法效率。
摘要：
本文提出了一种新的隐式重启广义二阶Arnoldi（GSOAR）方法的移位策略。在隐式重启过程中，通过执行p = m - k次隐式移位QR迭代，可以从m步GSOAR分解得到k步分解。隐式重启GSOAR方法的问题在于移位数量与子空间维度不匹配。对于p次QR迭代，有2p个移位。利用这些移位来过滤当前子空间中的冗余信息；使用更多移位可获得更好的更新子空间，但若使用超过p个移位，GSOAR分解的结构将被破坏。本文提出了一种新方法，能够利用所有2p个候选移位并保留特殊结构。该方法显著提升了算法的整体效率。数值实验展示了每个重启过程的效率。
英文摘要：
In this paper, a new shift strategy for the implicitly restarted generalized second-order Arnoldi (GSOAR) method is proposed. In implicitly restarted processes, we can get a $k$-step GSOAR decomposition from a $m$-step GSOAR decomposition by performing $p = m-k$ implicit shifted QR iterations. The problem of the implicitly restarted GSOAR is the mismatch between the number of shifts and the dimension of the subspace. There are $2p$ shifts for $p$ QR iterations. We use the shifts to filter out the unwanted information in the current subspace; when more shifts are used, one obtains a better updated subspace. But, if we use more than $p$ shifts, the structure of the GSOAR decomposition will be destroyed. We propose a novel method which can use all $2p$ candidates and preserve the special structure. The new method vastly enhances the overall efficiency of the algorithm. Numerical experiments illustrate the efficiency of every restart process.
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