A Nonlinear Generalization of the Bauer-Fike Theorem and Novel Iterative Methods for Solving Nonlinear Eigenvalue Problems

Nonlinear eigenvalue problems (NEPs) present significant challenges due to their inherent complexity and the limitations of traditional linear eigenvalue theory. This paper addresses these challenges by introducing a nonlinear generalization of the Bauer-Fike theorem, which serves as a foundational result in classical eigenvalue theory. This generalization provides a robust theoretical framework for understanding the sensitivity of eigenvalues in NEPs, extending the applicability of the Bauer-Fike theorem beyond linear cases. Building on this theoretical foundation, we propose novel iterative methods designed to efficiently solve NEPs. These methods leverage the generalized theorem to improve convergence rates and accuracy, making them particularly effective for complex NEPs with dense spectra. The adaptive contour integral method, in particular, is highlighted for its ability to identify multiple eigenvalues within a specified region of the complex plane, even in cases where eigenvalues are closely clustered. The efficacy of the proposed methods is demonstrated through a series of numerical experiments, which illustrate their superior performance compared to existing approaches. These results underscore the practical applicability of our methods in various scientific and engineering contexts. In conclusion, this paper represents a significant advancement in the study of NEPs by providing a unified theoretical framework and effective computational tools, thereby bridging the gap between theory and practice in the field of nonlinear eigenvalue problems.