The computation of eigenvalues and solutions of Mathieu's differential equation for noninteger order

ABSTRACT

Two algorithms for calculating the eigenvalues and solutions of Mathieu's differential equation for noninteger order are described. In the first algorithm, Leeb's method 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 1012. 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 1014, but has only been implemented for orders less than 10.5.

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INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Algorithm design and analysis

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Eigenvalues and eigenvectors (direct and iterative methods)

General Terms:
Algorithms, Performance

Keywords:
Floquet solutions, Mathieu equation, eigenvalues, noninteger order, numerical software, ordinary differential equations