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Numerical solution of nonlinear eigenvalue problems (NEPs) is frequentlyencountered in computational science and engineering. The applicability of mostexisting methods is limited by matrix structures, property of eigen-solutions,size of the problem, etc. This paper aims to break those limitations and todevelop robust and universal NEP solvers for large-scale engineeringapplications. The novelty lies in two aspects. First, a rational interpolationapproach (RIA) is proposed based on the Keldysh theorem for holomorphic matrixfunctions. Comparing with the existing contour integral approach (CIA), the RIAprovides the possibility to select sampling points in more general regions andhas advantages in improving accuracy and reducing computational cost. Second, aresolvent sampling scheme using the RIA is proposed for constructing reliablesearch spaces for the Rayleigh-Ritz procedure, based on which a robusteigen-solver, denoted by RSRR, is developed for solving general NEPs. RSRR canbe easily implemented and parallelized. The advantages of the RIA and theperformance of RSRR are demonstrated by a variety of benchmark and practicalproblems.

Solving large-scale nonlinear eigenvalue problems by rational interpolation approach and resolvent sampling based Rayleigh-Ritz method