{"title":"A Nonlinear Generalization of the Bauer-Fike Theorem and Novel Iterative Methods for Solving Nonlinear Eigenvalue Problems","authors":"Ronald Katende","doi":"arxiv-2409.11098","DOIUrl":null,"url":null,"abstract":"Nonlinear eigenvalue problems (NEPs) present significant challenges due to\ntheir inherent complexity and the limitations of traditional linear eigenvalue\ntheory. This paper addresses these challenges by introducing a nonlinear\ngeneralization of the Bauer-Fike theorem, which serves as a foundational result\nin classical eigenvalue theory. This generalization provides a robust\ntheoretical framework for understanding the sensitivity of eigenvalues in NEPs,\nextending the applicability of the Bauer-Fike theorem beyond linear cases.\nBuilding on this theoretical foundation, we propose novel iterative methods\ndesigned to efficiently solve NEPs. These methods leverage the generalized\ntheorem to improve convergence rates and accuracy, making them particularly\neffective for complex NEPs with dense spectra. The adaptive contour integral\nmethod, in particular, is highlighted for its ability to identify multiple\neigenvalues within a specified region of the complex plane, even in cases where\neigenvalues are closely clustered. The efficacy of the proposed methods is\ndemonstrated through a series of numerical experiments, which illustrate their\nsuperior performance compared to existing approaches. These results underscore\nthe practical applicability of our methods in various scientific and\nengineering contexts. In conclusion, this paper represents a significant\nadvancement in the study of NEPs by providing a unified theoretical framework\nand effective computational tools, thereby bridging the gap between theory and\npractice in the field of nonlinear eigenvalue problems.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"25 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11098","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
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.