Asymptotics of eigenvalues for pentadiagonal symmetric toeplitz matrices

Abstract

In this paper, we find uniform asymptotic formulas for all the eigenvalues of certain pentadiagonal symmetric Toeplitz matrices of large dimension. The entries of the matrices are real and we consider the case where the real-valued generating function has a minimum and a maximum such that its fourth derivative at the minimum and its second derivative at the maximum are nonzero. This is not the simple-loop case considered in [1] and [2]. We apply the main result of [7] and obtain nonlinear equations for the eigenvalues. It should be noted that our equations have a more complicated structure than the equations in [1] and [2]. Therefore, we required a more delicate method for its asymptotic analysis.