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The computation of eigenvalues and solutions of Mathieu’s differential equation for noninteger order. (English) Zbl 0892.65057

Summary: Two algorithms for calculating the eigenvalues and solutions of Mathieu’s differential equation for noninteger order are described. In the first algorithm, W. R. Leeb’s method [Algorithm 537: Characteristic values of Mathieu’s differential equation. ACM Trans. Math. Softw. 5, No. 1, 112-117 (1979)] is generalized, expanding the Mathieu equation in Fourier series and diagonalizing the symmetric tridiagonal matrix that results. Numerical testing was used to parameterize the minimum matrix dimension that must be used to achieve accuracy in the eigenvalue of one part in ${10}^{12}$. This method returns a set of eigenvalues below a given order and their associated solutions simultaneously.

A second algorithm is presented which uses approximations to the eigenvalues (Taylor series and asymptotic expansions) and then iteratively corrects the approximations using Newton’s method until the corrections are less than a given tolerance. A backward recursion of the continued fraction expansion is used. The second algorithm is faster and is optimized to obtain accuracy of one part in ${10}^{14}$, but has only been implemented for orders less than 10.5. For the algorithms see ibid. 19, No. 3, 391-406 (1993; reviewed below).

##### MSC:
 65L15 Eigenvalue problems for ODE (numerical methods) 34B30 Special ODE (Mathieu, Hill, Bessel, etc.) 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions (ODE)
MTIEU1