# The nonlinear eigenvalue problem *

@article{Gttel2017TheNE, title={The nonlinear eigenvalue problem *}, author={Stefan G{\"u}ttel and Françoise Tisseur}, journal={Acta Numerica}, year={2017}, volume={26}, 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

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#### References

SHOWING 1-10 OF 235 REFERENCES

A block Newton method for nonlinear eigenvalue problems

- 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

- Mathematics
- 2008

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 developed… Expand

Chebyshev interpolation for nonlinear eigenvalue problems

- Mathematics
- 2012

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 of… Expand

A projection method for nonlinear eigenvalue problems using contour integrals

- Computer Science, Mathematics
- JSIAM Lett.
- 2013

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 eigenvalue… Expand

Rational Krylov Methods for Nonlinear Eigenvalue Problems

- Computer Science
- 2013

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

- Mathematics
- 2004

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 the… Expand

A numerical method for nonlinear eigenvalue problems using contour integrals

- Computer Science, Mathematics
- JSIAM Lett.
- 2009

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

- Mathematics
- 2006

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 dependence… Expand

Solving Rational Eigenvalue Problems via Linearization

- Mathematics, Computer Science
- SIAM J. Matrix Anal. Appl.
- 2011

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

- Mathematics
- 2017

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 the… Expand