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

2015/12/28 by James Bremer, Bremer, James · 1 citation
Engineering · Mathematics · #Advanced Electrical Measurement Techniques #FOS: Mathematics #Mathematical functions and polynomials #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.1512.08357

openalex publication_date 2015/12/28 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

We describe a method for calculating the roots of special functions\nsatisfying second order linear ordinary differential equations. It exploits the\nrecent observation that the solutions of a large class of such equations can be\nrepresented via nonoscillatory phase functions, even in the high-frequency\nregime. Our algorithm achieves near machine precision accuracy and the time\nrequired to compute one root of a solution is independent of the frequency of\noscillations of that solution. Moreover, despite its great generality, our\napproach is competitive with specialized, state-of-the-art methods for the\nconstruction of Gaussian quadrature rules of large orders when it used in such\na capacity. The performance of the scheme is illustrated with several numerical\nexperiments and a Fortran implementation of our algorithm is available at the\nauthor's website.\n

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• Fast and accurate computation of classical Gaussian quadratures

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