Hamiltonian Polynomial Eigenvalue Problems

We present in this paper a new method for solving polynomial eigenvalue problem. We give methods that decompose a skew-Hamiltonian matrix using Cholesky like-decomposition. We transform first the polynomial eigenvalue problem to an equivalent skew-Hamiltonian/Hamiltonian pencil. This process is known as linearization. Decomposition of the skew-Hamiltonian matrix is the fundamental step to convert a structured polynomial eigenvalue problem into a standard Hamiltonian eigenproblem. Numerical examples are given.


Introduction
In this work we propose a numerical approach for solving the k th degree polynomial eigenvalue problem Problem (P) arises in many applications in science and engineering, ranging from the dynamical analysis of structural systems such as bridges and buildings to theories of elementary particles in atomic physics [1] [2]. It's also the most important task in the vibration analysis of buildings, machines, and vehicles [3].
We first transform our k th degree polynomial eigenvalue problem (P) to an  [ Proposition 2.2. The inverse of a regular upper J-triangular 2n-by-2n real matrix (respectively, lower J-triangular) is also upper J-triangular (respectively, also lower J-triangular).

Cholesky Like-Decomposition for Skew-Hamiltonian Matrix
In this section, we study different ways to compute J R R decomposition of a real skew-Hamiltonian matrix 2 2 n n M × ∈  . We began first by giving some interesting theoretical results.

Definition and Theoretical Results
Definition 3.1. The 2n-by-2n real skew-Hamiltonian matrix M is called and a 2n-by-2n real skew-Hamiltonian matrix M, Lemma 3.1. If M is a 2n-by-2n real skew-Hamiltonian and J-definite matrix, then M is invertible.
which is contradictory with the hypothesis.  Theorem 3.2. If M is a 2n-by-2n real skew-Hamiltonian, J-definite matrix, then M has an LU J-factorization.

Proof. Let
Proof. Since the matrix M is skew-Hamiltonian, then by taking U N = ∆ we

Method 1
We construct an algorithm that gives decomposition J R R of skew-Hamiltonian matrices via a LU J-decomposition. Proof. By corollary 3.3, where D is as given above, then

Method 2
We study now a method that constructs decomposition J R R of skew-Hamiltonian J-definite matrices.  The method yield the following algorithm.

Polynomial Eigenvalue Problems
Many applications give rise to structured matrix polynomial eigenvalue problems The numerical solution of this polynomial eigenvalue problem is one of the most important tasks in the vibration analysis of buildings, machines and vehicles [7]. In many applications, the coefficient matrices have particular structure and it is important that numerical methods respect this structure. A popular approach for solving the polynomial eigenvalue problem ( ) 0 trix H is then obtained by T T 1 J R AR − − . Eigenvalue problems of this type arise property that all eigenvalues appear in quadruples ( ) , , , λ λ λ λ − − , the spectrum is symmetric with respect to the real and imaginary axes.

Numerical Examples
We present computed eigenvalues that solve the k th degree polynomial eigenva- which is transforming to a standard eigenvalue problem of dimension kn kn × . We also compute the error consisting in ( ) ( ) We obtain a 144 × 144 quartic pencil, whose 576 eigenvalues are shown in Figure 1 given above.

Conclusion
We have proposed a numerical approach for solving polynomial eigenvalue problems structured. We first transform polynomial eigenvalue problem . Numerical methods based on these structured linearizations are expected to be more effective in computing accurate eigenvalues in practical applications. My future work based on this current study is to solve the large matrix equations applied in signal processing, image restoration and model reduction in control theory.