论文标题
对常规非线性特征值问题的Rayleigh-Ritz和完善的Rayleigh-Ritz方法的分析
An analysis of the Rayleigh-Ritz and refined Rayleigh-Ritz methods for regular nonlinear eigenvalue problems
论文作者
论文摘要
我们建立了瑞利 - 里兹方法的一般融合理论和用于计算一些简单的eigenpair $（λ_ {*}，x _ {*}）$的精制雷利 - 里茨方法，用于给定的分析性非线性非线性特征eigenvalue问题（NEP）。就$ x _ {*} $的偏差$ \ varepsilon $从给定的子空间$ \ mathcal {w} $中，我们建立了Ritz Value，Ritz Vector和Ritz Ritz Vector的先验收敛结果。结果表明，作为$ \ varepsilon \ rightarrow 0 $，存在一个无条件收敛到$λ_*$的Ritz值，相应的精制Ritz vector也这样做也这样做，但是Ritz Vector有条件地收敛，并且可能无法收敛甚至可能是唯一的。我们还根据给定近似特征台的可计算残差规范给出了近似特征向量的误差，并给出了精制的Ritz Vector和Ritz Vector的误差以及相应残留规范的误差。这些结果非试图扩展到这两种方法的线性特征值问题的一些收敛结果到NEP。构建示例以说明主要结果。
We establish a general convergence theory of the Rayleigh--Ritz method and the refined Rayleigh--Ritz method for computing some simple eigenpair $(λ_{*},x_{*})$ of a given analytic regular nonlinear eigenvalue problem (NEP). In terms of the deviation $\varepsilon$ of $x_{*}$ from a given subspace $\mathcal{W}$, we establish a priori convergence results on the Ritz value, the Ritz vector and the refined Ritz vector. The results show that, as $\varepsilon\rightarrow 0$, there exists a Ritz value that unconditionally converges to $λ_*$ and the corresponding refined Ritz vector does so too but the Ritz vector converges conditionally and it may fail to converge and even may not be unique. We also present an error bound for the approximate eigenvector in terms of the computable residual norm of a given approximate eigenpair, and give lower and upper bounds for the error of the refined Ritz vector and the Ritz vector as well as for that of the corresponding residual norms. These results nontrivially extend some convergence results on these two methods for the linear eigenvalue problem to the NEP. Examples are constructed to illustrate the main results.
