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A combined symbolic and numerical algorithm for the computation of zeros of orthogonal polynomials and special functions. (English) Zbl 1024.65039

Summary: 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 difference-differential 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.

MSC:

65H05 Numerical computation of solutions to single equations
33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
65D20 Computation of special functions and constants, construction of tables
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33F05 Numerical approximation and evaluation of special functions
68W30 Symbolic computation and algebraic computation

Software:

RFSFNS; Maple; GNOME; ELF; ALGOL 60
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Full Text: DOI

References:

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