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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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