Fast algorithms for spherical harmonic expansions, II
Journal of Computational Physics
Fast construction of Fejér and Clenshaw-Curtis rules for general weight functions
Computers & Mathematics with Applications
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We describe a procedure for the determination of the roots of functions satisfying second-order ordinary differential equations, including the classical special functions. The scheme is based on a combination of the Prüfer transform with the classical Taylor series method for the solution of ordinary differential equations and requires $O(1)$ operations for the determination of each root. When the functions in question are classical orthogonal polynomials (Legendre, Hermite, etc.), the techniques presented here also provide tools for the evaluation of the weights for the associated Gaussian quadratures. The performance of the scheme for several classical special functions (prolate spheroidal wave functions, Bessel functions, and Legendre, Hermite, and Laguerre polynomials) is illustrated with numerical examples.