THE COMPUTATION OF EIGENVALUES AND SOLUTIONS OF MATHIEU DIFFERENTIAL-EQUATION FOR NONINTEGER ORDER

Authors
Citation
Rb. Shirts, THE COMPUTATION OF EIGENVALUES AND SOLUTIONS OF MATHIEU DIFFERENTIAL-EQUATION FOR NONINTEGER ORDER, ACM transactions on mathematical software, 19(3), 1993, pp. 377-390
Citations number
40
Language
INGLESE
art.tipo
Article
Categorie Soggetti
Computer Sciences",Mathematics
ISSN journal
0098-3500
Volume
19
Issue
3
Year of publication
1993
Pages
377 - 390
Database
ISI
SICI code
0098-3500(1993)19:3<377:TCOEAS>2.0.ZU;2-Y
Abstract
Two algorithms for calculating the eigenvalues and solutions of Mathie u's differential equation for noninteger order are described. In the f irst algorithm, Leeb's method is generalized, expanding the Mathieu eq uation in Fourier series and diagonalizing the symmetric tridiagonal m atrix that results. Numerical testing was used to parameterize the min imum matrix dimension that must be used to achieve accuracy in the eig envalue of one part in 10(12). This method returns a set of eigenvalue s below a given order and their associated solutions simultaneously. A second algorithm is presented which uses approximations to the eigenv alues (Taylor series and asymptotic expansions) and then iteratively c orrects the approximations using Newton's method until the corrections are less than a given tolerance. A backward recursion of the continue d fraction expansion is used. The second algorithm is faster and is op timized to obtain accuracy of one part in 10(14), but has only been im plemented for orders less than 10.5.