Peculiarities of calculation of the roots of the Mathieu equation in the Maple system

When studying various parameters and characteristics of the propagation of electromagnetic waves in gyrotropic elliptical waveguides, the dispersion equations that are solutions of the Helmholtz equations contain various Mathieu functions and their derivatives. Therefore, when solving the corresponding dispersion equations, it is usually necessary to numerically calculate the roots of the Mathieu function. To do this, it is possible to use the Maple computer mathematics system, which has a built-in command (function) for numerically calculating the roots of the Mathieu function. However, the direct use of this command for this purpose causes certain difficulties. This article analyzes these difficulties and features of calculating the roots of the Mathieu equation in the Maple system. The main disadvantage of the built-in command for the numerical calculation of all roots of the Mathieu equation that exist on a given interval is revealed. A recursive algorithm and a corresponding software implementation of the algorithm in the Maple system are proposed, which eliminate the noted drawback. The proposed algorithm and software implementation have been tested by comparing the results of calculating the roots of the Mathieu function with known roots from a reference book on special functions.