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

Numerical solution of nonlinear eigenvalue problems (NEPs) is frequently encountered in computational science and engineering. The applicability of most existing methods is limited by the matrix structures, properties of the eigen-solutions, sizes of the problems, etc. This paper aims to remove those limitations and develop robust and universal NEP solvers for large-scale engineering applications. The novelty lies in two aspects. First, a rational interpolation approach (RIA) is proposed based on the Keldysh theorem for holomorphic matrix functions. Comparing with the existing contour integral approach, the RIA provides the possibility to select sampling points in more general regions and has advantages in improving the accuracy and reducing the computational cost. Second, a resolvent sampling scheme using the RIA is proposed to construct reliable search spaces for the Rayleigh–Ritz procedure, based on which a robust eigen-solver, called resolvent sampling based Rayleigh–Ritz method (RSRR), is developed for solving general NEPs. The RSRR can be easily implemented and parallelized. The advantages of the RIA and the performance of the RSRR are demonstrated by a variety of benchmark and application examples.