Quadratic eigenvalue problem: Difference between revisions

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In [[mathematics]], the '''quadratic eigenvalue problem<ref>F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM

Rev., 43 (2001), pp. 235–286.</ref> (QEP)''', is to find [[scalar (mathematics)|scalar]] [[eigenvalue]]s <math>\lambda</math>, left [[eigenvector]]s <math>y</math> and right eigenvectors <math>x</math> such that

:<math> Q(\lambda)x = 0\text{ and }y^\ast Q(\lambda) = 0,</math>

where <math>Q(\lambda)=\lambda^2 A_2 + \lambda A_1 + A_0</math>, with matrix coefficients <math>A_2, \, A_1, A_0 \in \mathbb{C}^{n \times n}</math> and we require that <math>A_2\,\neq 0</math>, (so that we have a nonzero leading coefficient). There are <math>2n</math> eigenvalues that may be ''infinite'' or finite, and possibly zero. This is a special case of a [[nonlinear eigenproblem]]. <math>Q(\lambda)</math> is also known as a quadratic matrix [[polynomial matrix]].

==Applications==

A QEP can result in part of the dynamic analysis of structures [[Discretization|discretized]] by the [[finite element method]]. In this case the quadratic, <math>Q(\lambda)</math> has the form <math>Q(\lambda)=\lambda^2 M + \lambda C + K</math>, where <math>M</math> is the [[mass matrix]], <math>C</math> is the [[damping matrix]] and <math>K</math> is the [[stiffness matrix]].

Other applications include vibro-acoustics and fluid dynamics.

==Methods of solution==

Direct methods for solving the standard or [[Generalized eigenvalue problem|generalized eigenvalue problems]] <math> Ax = \lambda x</math> and <math> Ax = \lambda B x </math>

are based on transforming the problem to [[Schur form|Schur]] or [[Schur decomposition#Generalized Schur decomposition|Generalized Schur]] form. However, there is no analogous form for quadratic matrix polynomials.

One approach is to transform the quadratic matrix polynomial to a linear [[matrix pencil]] (<math> A-\lambda B</math>), and solve a generalized

eigenvalue problem. Once eigenvalues and eigenvectors of the linear problem have been determined, eigenvectors and eigenvalues of the quadratic can be determined.

The most common linearization is the first companion[[Companion matrix|companion]] linearization

:<math>

L(\lambda) =