On the double points of a Mathieu equation

For a Mathieu equation with parameter q, the eigenvalues can be regarded as functions of the variable q. Our aim is to find q when adjacent eigenvalues of the same type become equal yielding double points of the given Mathieu equation. The problem reduces to an equivalent eigenvalue problem of the form BX=λX, where B is an infinite tridiagonal matrix. A method is developed to locate the first double eigenvalue to any required degree of accuracy when q is an imaginary number. Computational results are given to illustrate the theory for the first double eigenvalue. Numerical results are given for some subsequent double points.