The Quadratic Eigenvalue Problem

. We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skew-Hermitian) and the spectral properties of the problem. We classify numerical methods and catalogue available software. Key words. quadratic eigenvalue problem, eigenvalue, eigenvector, -matrix, matrix polynomial, second-order differential equation, vibration, Millennium footbridge, overdamped system, gyroscopic system, linearization, backward error, pseudospectrum, condition number, Krylov methods, Arnoldi method, Lanczos method, Jacobi-Davidson method AMS subject classifications. 65F30 Contents 1 Introduction 2 2 Applications of QEPs 4 2.1 Second-order differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Vibration analysis of structural systems --...