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

Journal title

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.