We analyze the Rayleigh-Ritz method and the refined Rayleigh-Ritz method for computing an approximation of a simple eigenpair ($\lambda_{*},x_{*}$) of a given nonlinear eigenvalue problem. For a given subspace $\mathcal{W}$ that contains a sufficiently accurate approximation to $x_{*}$, we establish convergence results on the Ritz value, the Ritz vector and the refined Ritz vector as the deviation $\varepsilon$ of $x_{*}$ from $\mathcal{W}$ approaches zero. We also derive 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 extend the convergence results of these two methods for the linear eigenvalue problem to the nonlinear case. We construct examples to illustrate some of the results.

翻译：我们分析了Rayleigh-Ritz的方法和改良的Rayleigh-Ritz方法,以计算一个简单的egenpair(lambda ⁇,x ⁇ $)的非线性亚值问题的近似值。对于含有足够精确近近似值的某个子空间$\mathcal{W}美元,我们将Ritz值、Ritz矢量和改良的Ritz矢量的趋同结果确定为从$\mathcal{W}美元到0美元的偏差$\varepsilon。我们还得出了精化的Ritz矢量和Ritz矢量的差错以及相应的剩余规范的上下限。~这些结果将这两种方法对线性亚值问题的趋同结果延伸至非线性案例。我们用实例来说明其中的一些结果。