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You may be interested in "On the Numerical Calculation of the Roots of Special Functions Satisfying Second Order Ordinary Differential Equations", Bremer,


This is a classical problem in numerical methods research: evaluating the zeros of special functions. Many years of research have gone into devising efficient methods. The canonical starting point for anything related to this is the classic book by Abramowicz and Stegun. Scanned versions of the book can easily be found on the internet (I'm not sure if these ...

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