Quadratic eigenvalue problem: Difference between revisions

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==Spectral theory==

A QEP is said to be <em>regular</em> if <math>\text{Detdet} (Q(\lambda)) \not \equiv 0</math> identically. The coefficient of the <math>\lambda^{2n}</math> term in <math>\text{Detdet}(Q(\lambda))</math> is <math>\text{Detdet}(M)</math>, implying that the QEP is regular if <math>M</math> is nonsingular.

Eigenvalues at infinity and eigenvalues at 0 may be exchanged by considering the reversed polynomial, <math> \lambda^2 Q(\lambda^{-1}) = \lambda^2 K + \lambda C + M </math>. As there are <math> 2n</math> eigenvectors in a <math>n</math> dimensional space, the eigenvectors cannot be orthogonal. It is possible to have the same eigenvector attached to different eigenvalues.

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==Applications==

=== Systems of differential equations ===

Quadratic eigenvalue problems arise naturally in the solution of systems of second order [[Linear differential equation|linear differential equations]] without forcing:

:<math> M q''(t) +C q'(t) + K q(t) = 0 </math>

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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==

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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.

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\end{bmatrix}.

</math>

For convenience, one often takes <math>N</math> theto be the <math>n\times n</math> [[identity matrix]]. We solve <math> L(\lambda) z = 0 </math> for <math> \lambda </math> and <math>z</math>, for example by computing the Generalized Schur form. We can then

take the first <math>n</math> components of <math>z</math> as the eigenvector <math>x</math> of the original quadratic <math>Q(\lambda)</math>.