Computation features for Mathieu functions of arbitrary orders

An algorithm for successive computation of wide-ranging order Mathieu functions (from 0 to several hundreds) is proposed. The algorithm is based on approximation by Fourier series with beforehand accuracy. It is shown, that the increasing in order of Mathieu function up to hundreds and more requires the high precision calculation of the continued fraction consisting of Fourier coefficients. The criteria for evaluating of accuracy are introduced. Perspective of using of high-order Mathieu function approximation for single high-frequency harmonic is discussed. Numerical examples are given.