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We establish a general convergence theory of the Rayleigh--Ritz method andthe 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, theRitz vector and the refined Ritz vector. The results show that, as$\varepsilon\rightarrow 0$, there exists a Ritz value that unconditionallyconverges to $\lambda_*$ and the corresponding refined Ritz vector does so toobut the Ritz vector converges conditionally and it may fail to converge andeven may not be unique. We also present an error bound for the approximateeigenvector in terms of the computable residual norm of a given approximateeigenpair, and give lower and upper bounds for the error of the refined Ritzvector and the Ritz vector as well as for that of the corresponding residualnorms. These results nontrivially extend some convergence results on these twomethods for the linear eigenvalue problem to the NEP. Examples are constructedto illustrate the main results.

An analysis of the Rayleigh-Ritz and refined Rayleigh-Ritz methods for regular nonlinear eigenvalue problems