An Analysis of the Rayleigh–Ritz and Refined Rayleigh–Ritz Methods for Regular Nonlinear Eigenvalue Problems
Zhongxiao Jia, Qingqing Zheng · SIAM Journal on Matrix Analysis and Applications · 2025
Abstract. We establish a general convergence theory of the Rayleigh–Ritz method and the refined Rayleigh–Ritz method for computing some simple eigenpair [Formula: see text] of a given analytic regular nonlinear eigenvalue problem (NEP). In terms of the deviation [Formula: see text] of [Formula: see text] from a given subspace [Formula: see text], we establish a priori convergence results on the Ritz value, the Ritz vector, and the refined Ritz vector. The results show that, as [Formula: see text], there exists a Ritz value that unconditionally converges to [Formula: see text], as does the corresponding refined Ritz vector, but the Ritz vector converges conditionally and 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.