抄録
Given a complex number λ, we consider the problem of finding those values of q for which the Mathieu's equation ω″(z)+(λ-2qcos2z)ω(z)=0 admits π- or 2π- periodic solutions. This is an inverse problem to the usual one where q is given and λ, an eigenvalue of the equation, is unknown. In this paper, we propose to solve the inverse problem by a matrix method. We will give an extremely accurate asymptotic error estimate. In addition, we present a method with a good rate of convergence that calculates the point (q, λ) such that λ is an eigenvalue satisfying dλ/dq=0.
