The nonlinear eigenvalue problem *

  title={The nonlinear eigenvalue problem *},
  author={Stefan G{\"u}ttel and Françoise Tisseur},
  journal={Acta Numerica},
  pages={1 - 94}
Nonlinear eigenvalue problems arise in a variety of science and engineering applications, and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques. Solvers based on Newton’s method, contour… Expand
A Riesz-projection-based method for nonlinear eigenvalue problems
An algorithm for general nonlinear eigenvalue problems to compute eigenvalues within a chosen contour and to compute the corresponding eigenvectors is proposed and applied to two examples from physics. Expand
Automatic rational approximation and linearization of nonlinear eigenvalue problems
We present a method for solving nonlinear eigenvalue problems using rational approximation. The method uses the AAA method by Nakatsukasa, S\`{e}te, and Trefethen to approximate the nonlinearExpand
A nonlinear eigenmode solver for linear viscoelastic structures
The solution method for the considered nonlinear eigenvalue problem is based on the contour integral method, where special focus is put on the efficient numerical computation of the linear system along the boundary of the given search area. Expand
A rational approximation method for solving acoustic nonlinear eigenvalue problems
We present two approximation methods for computing eigenfrequencies and eigenmodes of large-scale nonlinear eigenvalue problems resulting from boundary element method (BEM) solutions of some types ofExpand
FEAST Eigensolver for Nonlinear Eigenvalue Problems
The nonlinear FEAST algorithm can be used to solve nonlinear eigenvalue problems for the eigenpairs whose eigenvalues lie in a user-defined region in the complex plane, thereby allowing for the calculation of large numbers of eigenPairs in parallel. Expand
The subspace iteration method for nonlinear eigenvalue problems occurring in the dynamics of structures with viscoelastic elements
Abstract The paper presents an extension of the subspace iteration method for application in systems with viscoelastic damping elements, which are described by both classical and fractional models.Expand
Nonlinearization of two-parameter eigenvalue problems
We investigate a technique to transform a linear two-parameter eigenvalue problem, into a nonlinear eigenvalue problem (NEP). The transformation stems from an elimination of one of the equations inExpand
What Do You Mean by "Nonlinear Eigenvalue Problems"?
This review paper tries to collect some points of possible common interest for both fields: the nonlinear eigenvalue problem and the research in this area seems to follow two quite different directions. Expand
Nonlinearizing two-parameter eigenvalue problems
By exploiting the structure of the NEP, the technique to transform a linear two-parameter eigen value problem, into a nonlinear eigenvalue problem (NEP) is investigated, which allows general solution methods for NEPs to be directly applied. Expand
Perturbation theory of nonlinear, non-self-adjoint eigenvalue problems: Simple eigenvalues
Abstract The study of the vibrational modes and stability of a given physical system is strongly tied to the efficient numerical evaluation of its eigenvalues. The operators governing theExpand


A block Newton method for nonlinear eigenvalue problems
  • D. Kressner
  • Mathematics, Computer Science
  • Numerische Mathematik
  • 2009
The purpose of this work is to show that the concept of invariant pairs offers a way of representing eigenvalues and eigenvectors that is insensitive to this phenomenon, and to demonstrate the use of this concept in the development of numerical methods, a novel block Newton method is developed. Expand
Nonlinear Eigenvalue Problems: Newton-type Methods and Nonlinear Rayleigh Functionals
Nonlinear eigenvalue problems arise in many fields of natural and engineering sciences. Theoretical and practical results are scattered in the literature and in most cases they have been developedExpand
Chebyshev interpolation for nonlinear eigenvalue problems
This work is concerned with numerical methods for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. In particular, we focus on eigenvalue problems for which the evaluation ofExpand
A projection method for nonlinear eigenvalue problems using contour integrals
In this paper, we indicate that the Sakurai-Sugiura method with Rayleigh-Ritz projection technique, a numerical method for generalized eigenvalue problems, can be extended to nonlinear eigenvalueExpand
Rational Krylov Methods for Nonlinear Eigenvalue Problems
The Compact Rational Krylov (CORK) method is proposed as a generic class of numerical methods for solving nonlinear eigenvalue problems and is able to solve problems of high dimension and high degree in an efficient and reliable way. Expand
Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods
We discuss the state of the art in numerical solution methods for large scale polynomial or rational eigenvalue problems. We present the currently available solution methods such as theExpand
A numerical method for nonlinear eigenvalue problems using contour integrals
A contour integral method is proposed to solve nonlinear eigenvalue problems numerically by reducing the original problem to a linear eigen value problem that has identical eigenvalues in the domain. Expand
Preconditioned iterative methods for a class of nonlinear eigenvalue problems
Abstract This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependenceExpand
Solving Rational Eigenvalue Problems via Linearization
It is shown that solving a class of rational eigen value problems is just as convenient and efficient as solving linear eigenvalue problems. Expand
Solving large‐scale nonlinear eigenvalue problems by rational interpolation and resolvent sampling based Rayleigh–Ritz method
Summary Numerical solution of nonlinear eigenvalue problems (NEPs) is frequently encountered in computational science and engineering. The applicability of most existing methods is limited by theExpand