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 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 $\lambda_*$ 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.

翻译：针对解析正则非线性特征值问题（NEP）中特定简单特征对$(\lambda_{*},x_{*})$的计算，我们建立了Rayleigh-Ritz方法及其改进形式的一般收敛理论。通过特征向量$x_{*}$与给定子空间$\mathcal{W}$的偏离量$\varepsilon$，我们获得了关于Ritz值、Ritz向量及改进Ritz向量的先验收敛结果。研究表明：当$\varepsilon\rightarrow 0$时，必存在无条件收敛至$\lambda_*$的Ritz值，其对应的改进Ritz向量同样无条件收敛；而Ritz向量仅具备条件收敛性，可能无法收敛甚至不具唯一性。我们进一步建立了基于可计算残差范数的近似特征向量误差界，给出了改进Ritz向量与Ritz向量的误差上下界及其对应残差范数的界值。这些结果将线性特征值问题中两种方法的收敛理论非平凡地推广至NEP情形。文中构造了若干算例以佐证主要结论。