Refined Jacobi-Davidson Type Method For A Right Definite Two-Parameter Eigenvalue Problem

In this paper,we discusses the following form of the so-called two-parameter eigenvalue problem:A₁x₁=λB₁x₁+μC₁x₁,A₂x₂=λB₂x₂+μC₂x₂.where A_i,B_i and C_i are given n_i×n_i matrices,x_i are n_i vectors for i=1,2.,λandμare scalars.A pair(λ,μ) is called an eigenvalue if it satisfies the above equation for nonzero vectors x₁,x₂.Then the tensor product x₁(?) x₂ is called the corresponding eigenvector.Two-parameter eigenvalue problems of this kind arise in a variety of applications.A Jacobi-Davidson type method is proposed in for a right definite twoparameter eigenvalue problem.In their paper,M.E.Hochstenbach and B. Plestenjak considered that the refined method is not suitable for two-parameter eigenvalue problems because of high costs for computation,poor convergence of refined Ritz vectors and incapacity for computing more than one eigenvalue.In this paper,we show that it is not the case.For a right definite two-parameter eigenvalue problem,We propose an efficient refined Jacobi-Davidson type method and show that refined Ritz vectors have better convergence than Ritz vectors and (refined) Ritz values is convergent.The paper has been organized as follows.In Chapter 1,we give a introduction of two-parameter eigenvalue problems and its background.In Chapter 2,we review the Jacobi-Davidson method in.In Chapter 3,we prove the convergence of Ritz values(Theorem 4) and illustrate that the convergence of refined Ritz vectors is better than Ritz vectors.Our algorithm is proposed in Chapter 4.We conclude with experiments and a conclusion in Section 5.