Spectral collocation solutions to multiparameter Mathieu's system

Our main aim is the accurate computation of a large number of specified eigenvalues and eigenvectors of Mathieu's system as a multiparameter eigenvalue problem (MEP). The reduced wave equation, for small deflections, is solved directly without approximations introduced by the classical Mathieu functions. We show how for moderate values of the cut-off collocation parameter the QR algorithm and the Arnoldi method may be applied successfully, while for larger values the Jacobi-Davidson method is the method of choice with respect to convergence, accuracy and memory usage.