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Description:
In mathematics, the quadratic eigenvalue problem (QEP), is to find scalar eigenvalues [itex]lambda,[/itex], left eigenvectors [itex]y,[/itex] and right eigenvectors [itex]x,[/itex] such that

[itex] Q(lambda)x = 0texty^ast Q(lambda) = 0,, [/itex]

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

## Applications

A QEP can result in part of the dynamic analysis of structures discretized by the finite element method. In this case the quadratic, [itex]Q(lambda),[/itex] has the form [itex]Q(lambda)=lambda^2 M + lambda C + K,[/itex], where [itex]M,[/itex] is the mass matrix, [itex]C,[/itex] is the damping matrix and [itex]K,[/itex] is the stiffness matrix.Other applications include vibro-acoustics and fluid dynamics.

## Methods of Solution

Direct methods for solving the standard or...

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