A polynomial(high-order) eigenvalue problem is a variation of the standard eigenvalue problem, where the equation $Ax=\lambda x$ is replaced by a polynomial equation of the form

A_0 +\lambda A_1+\lambda^2A_2+ \cdot \cdot \cdot +\lambda^nA_n=0

where $A_0,A_1,…,A_n$ are matrices and n is an integer greater than 2. The problem is to find the scalars $\lambda$ and non-zero vectors $x$ that are solutions of this equation.

The solutions of the high-order eigenvalue problem are not necessarily scalars, they can be complex numbers as well.

Polynomial eigenvalue problems have many important applications in physics, engineering, and other fields. In quantum mechanics, they are used to study the dynamics of systems with more than two degrees of freedom, such as the vibrations of a polyatomic molecule. In control theory, they are used to study the stability and control of systems with multiple inputs and outputs. In image processing, they are used to study the properties of images, such as their symmetry and smoothness.

Solving a high-order eigenvalue problem can be difficult, as it is a nonlinear problem. There are several methods to solve high-order eigenvalue problems, such as the polynomial eigenvalue method, the polynomial Lanczos method, and the polynomial Arnoldi method. Each method has its own advantages and disadvantages, and the choice of method depends on the specific problem and the available computational resources.

Polynomial Eigenvalue Problem and Viscoelasticity of Solid Mechanics

Viscoelasticity is a branch of continuum mechanics that deals with the behavior of materials that exhibit both viscous and elastic behavior. These materials can be modeled using high-order differential equations, which can lead to high-order eigenvalue problems when finding the natural frequencies and modes of vibration of the material.

For example, in viscoelasticity, the material’s response to a dynamic loading can be modeled using the Kelvin-Voigt model. The model is based on a set of differential equations that describe the behavior of the material in time. The solution of the differential equation lead to a high order eigenvalue problem where the matrix is of order 3.

Another example is the Maxwell model, which is also used to model viscoelastic materials. The model is based on a set of differential equations that describes the behavior of the material in time. The solution of the differential equation lead to a high order eigenvalue problem where the matrix is of order 4.

In summary, high-order eigenvalue problems are encountered in viscoelasticity, which is the study of materials that exhibit both viscous and elastic behavior. These materials are modeled using high-order differential equations, which leads to high-order eigenvalue problems when finding the natural frequencies and modes of vibration of the material.