We present a collection of 52 nonlinear eigenvalue problems in the form of a MATLAB toolbox. The collection contains problems from models of real-life applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial eigenvalue problems according to their structural properties. Identifiers based on these and other properties can be used to extract particular types of problems from the collection. A brief description of each problem is given. NLEVP serves both to illustrate the tremendous variety of applications of nonlinear eigenvalue problems and to provide representative problems for testing, tuning, and benchmarking of algorithms and codes.

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NLEVP: A Collection of Nonlinear Eigenvalue Problems

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Design Sensitivity Analysis Of Structural Systems

The solution of the equation XA + AXT = 0 and its application to the theory of orbits

Partial eigenstructure assignment for undamped vibration systems using acceleration and displacement feedback

https://hal.archives-ouvertes.fr/hal-01188551/document

A wave-based model reduction technique for the description of the dynamic behavior of periodic structures involving arbitrary-shaped substructures and large-sized finite element models

In this paper, based on Patel's algorithm (1993), we propose a structure-preserving algorithm for solving palindromic quadratic eigenvalue problems (QEPs). We also show the relationship between the structure-preserving algorithm and the URV-based structure-preserving algorithm by Schroder (2007). For large sparse palindromic QEPs, we develop a generalized $\top$-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi algorithm for solving the resulting $\top$-skew-Hamiltonian pencils. Numerical experiments show that our proposed structure-preserving algorithms perform well on the palindromic QEP arising from a finite element model of high-speed trains and rails.

https://ir.nctu.edu.tw:443/bitstream/11536/9855/1/000263103700017.pdf

Structure-Preserving Algorithms for Palindromic Quadratic Eigenvalue Problems Arising from Vibration of Fast Trains

Robust partial pole assignment problem for high order control systems

We consider quadratic eigenvalue problems in which the coefficient matrices, and hence the eigenvalues and eigenvectors, are functions of a real parameter. Our interest is in cases in which these functions remain differentiable when eigenvalues coincide. Many papers have been devoted to numerical methods for computing derivatives of eigenvalues and eigenvectors, but most require the eigenvalues to be well separated. The few that consider close or repeated eigenvalues place severe restrictions on the eigenvalue derivatives. We propose, analyze, and test new algorithms for computing first and higher order derivatives of eigenvalues and eigenvectors that are valid much more generally. Numerical results confirm the effectiveness of our methods for tightly clustered eigenvalues.

Computing Derivatives of Repeated Eigenvalues and Corresponding Eigenvectors of Quadratic Eigenvalue Problems

Quadratic inverse eigenvalue problem for damped gyroscopic systems

In this paper, we consider the minimum norm and robust partial quadratic eigenvalue assignment problems (PQEVAP). A complete theory on the existence of solutions for the PQEVAP is established. It is shown that solving the PQEVAP is essentially solving an eigenvalue assignment for a linear system of a much lower order, and the minimum norm and robust PQEVAPs are then concerning the minimum norm and robust eigenvalue assignment problems associated with this linear system. Based on this theory, an algorithm for solving the minimum norm and robust PQEVAPs is proposed, and its efficient behaviors are illustrated by some numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.

Jiang Qian

Papers

Structure-Preserving Algorithms for Palindromic Quadratic Eigenvalue Problems Arising from Vibration of Fast Trains

Robust partial pole assignment problem for high order control systems

Computing Derivatives of Repeated Eigenvalues and Corresponding Eigenvectors of Quadratic Eigenvalue Problems

Quadratic inverse eigenvalue problem for damped gyroscopic systems

The formulation and numerical method for partial quadratic eigenvalue assignment problems