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

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.