On Nonlinearization of 3-parameter Eigenvalue Problems

Abstract

In this paper, the linear 3-parameter eigenvalue problem (3PEP) in terms of matrix equations is considered. Using the Rayleigh Quotient iteration method, any one of the three parameters can be fixed to transform the problem into a linear 2-parameter eigenvalue problem (2PEP). This admits a family nonlinear eigenvalue problems (NEP) in one parameter. The transformation results from the elimination of the second equation of respective 2PEP, which is re-arranged as a generalized eigenvalue problem (GEP) of the form \(Ey=\lambda Fy\), where \(E\) and \(F\) are matrices \(n\times n\) over \(\mathbb{C}\), \(y\in \mathbb{C}^n\) is a non-zero vector and \(\lambda\) is a scalar. A review of some results of the condition number of NEP is also presented in this paper.