mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the quadratic eigenvalue problemF. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), pp. 235–286. (QEP), is to find

scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

\lambda

, left

eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

y

and right eigenvectors

x

such that :

Q(\lambda)x = 0 ~ \text ~ y^\ast Q(\lambda) = 0,

where

Q(\lambda)=\lambda^2 A_2 + \lambda A_1 + A_0

, with matrix coefficients

A_2, \, A_1, A_0 \in \mathbb^

and we require that

A_2\,\neq 0

, (so that we have a nonzero leading coefficient). There are

2n

eigenvalues that may be ''infinite'' or finite, and possibly zero. This is a special case of a

nonlinear eigenproblem In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form : ...

Q(\lambda)

is also known as a quadratic

polynomial matrix In mathematics, a polynomial matrix or matrix of polynomials is a matrix whose elements are univariate or multivariate polynomials. Equivalently, a polynomial matrix is a polynomial whose coefficients are matrices. A univariate polynomial matrix ' ...

Applications

A QEP can result in part of the dynamic analysis of structures discretized by the

finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...

. In this case the quadratic,

Q(\lambda)

has the form

Q(\lambda)=\lambda^2 M + \lambda C + K

, where

M

is the

mass matrix In analytical mechanics, the mass matrix is a symmetric matrix that expresses the connection between the time derivative \mathbf\dot q of the generalized coordinate vector of a system and the kinetic energy of that system, by the equation :T ...

C

is the damping matrix and

K

is the

stiffness matrix In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix is a matrix that represents the system of linear equations that must be solved in order to ascertain an approximate solution ...

. Other applications include vibro-acoustics and fluid dynamics.

Methods of solution

Direct methods for solving the standard or generalized eigenvalue problems

Ax = \lambda  x

and

Ax = \lambda B x

are based on transforming the problem to Schur or 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 In linear algebra, if A_0, A_1,\dots,A_\ell are n\times n complex matrices for some nonnegative integer \ell, and A_\ell \ne 0 (the zero matrix), then the matrix pencil of degree \ell is the matrix-valued function defined on the complex numbers L(\ ...

(

A-\lambda B

), 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 linearization :

L(\lambda) = 
\lambda
\begin
M & 0 \\
0 & I_n 
\end
+
\begin
C & K \\
-I_n & 0 
\end,

where

I_n

is the

n

-by-

n

identity matrix, with corresponding eigenvector :

z = 
\begin
\lambda x \\
x
\end.

We solve

L(\lambda) z = 0

for

\lambda

and

z

, for example by computing the Generalized Schur form. We can then take the first

n

components of

z

as the eigenvector

x

of the original quadratic

Q(\lambda)

. {{mathapplied-stub

References

Linear algebra