We establish a general convergence theory of the Rayleigh--Ritz method and the refined Rayleigh--Ritz method for computing some simple eigenpair ($\lambda_{*},x_{*}$) of a given analytic 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, and present sufficient convergence conditions for them. The results show that, as $\varepsilon\rightarrow 0$, there is a Ritz value that unconditionally converges to $\lambda_*$ and the corresponding refined Ritz vector does so too but the Ritz vector 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 some of the results.

翻译：本文针对一般解析非线性特征值问题（NEP）中简单特征对（λ_*, x_*）的计算，建立了瑞利-里兹方法与精细化瑞利-里兹方法的统一收敛理论。基于x_*偏离给定子空间𝒲的偏差ε，我们给出了瑞利值、瑞利向量及精细化瑞利向量的先验收敛结果，并提出了其充分收敛条件。结果表明：当ε→0时，存在一个无条件收敛至λ_*的瑞利值，且相应的精细化瑞利向量亦收敛，但瑞利向量可能不收敛甚至不唯一。进一步地，我们利用给定近似特征对的可计算残差范数，建立了近似特征向量的误差界，并给出了精细化瑞利向量与瑞利向量的误差下界与上界，以及相应残差范数的界。这些结果将线性特征值问题中关于这两种方法的收敛性结论非平凡地推广至NEP。文中构造了算例以说明部分结论。