On the numerical calculation of the roots of special functions satisfying second order ordinary differential equations

James Bremer

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arXiv · 1512.08357

On the numerical calculation of the roots of special functions satisfying second order ordinary differential equations

James Bremer

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Abstract

We describe a method for calculating the roots of special functions satisfying second order linear ordinary differential equations. It exploits the recent observation that the solutions of a large class of such equations can be represented via nonoscillatory phase functions, even in the high-frequency regime. Our algorithm achieves near machine precision accuracy and the time required to compute one root of a solution is independent of the frequency of oscillations of that solution. Moreover, despite its great generality, our approach is competitive with specialized, state-of-the-art methods for the construction of Gaussian quadrature rules of large orders when it used in such a capacity. The performance of the scheme is illustrated with several numerical experiments and a Fortran implementation of our algorithm is available at the author's website.

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James Bremer. 2015-12-28. On the numerical calculation of the roots of special functions satisfying second order ordinary differential equations. https://arxiv.org/abs/1512.08357

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