A combined symbolic and numerical algorithm for the computation of zeros of orthogonal polynomials and special functions

Authors:
Amparo Gil;Javier Segura
Affiliations:
Departmento de Matemáticas, Facultad Ciencias, Universidad Autónoma de Madrid, 28049-Madrid, Spain and Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911-Legan ...;Departamento de Matematicas, Estadistica y Computacion, Universidad de Cantabria, 39005 Santander, Spain
Venue:
Journal of Symbolic Computation - Special issue: Algebra and computer analysis
Year:
2003

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Abstract

A Maple algorithm for the computation of the zeros of orthogonal polynomials (OPs) and special functions (SFs) in a given interval [ x 1 , x 2 ] is presented. The program combines symbolic and numerical calculations and it is based on fixed point iterations. The program uses as inputs the analytic expressions for the coefficients of the three-term recurrence relation and a differencedifferential relation satisfied by the set of OPs or SFs. The performance of the method is illustrated with several examples: Hermite, Chebyshev, Legendre, Jacobi and Gegenbauer polynomials, Bessel, Coulomb and Conical functions.