Re: [Help-gsl] Hankel transform and Bessel functions zeros

[M J Carley]

On Tue, 20 Sep 2011, Brian Gough wrote:

> At Fri, 26 Aug 2011 11:28:14 +0100 (BST),
> Michael Carley wrote:
> > It seems to me that it might be useful to include an implementation of the
> > root finding algorithm in GSL since it could be used for a wide range of
> > special functions, not just Bessel functions, and would also improve the
> > accuracy of the Hankel transform, and I would be happy to recode my
> > implementation to GSL standards or pass it on to someone else who could do
> > so.
>
> Thanks for the email, and sorry for the delay in replying.  Can you
> give me an idea of how much of a drop-in replacement this would be --
> is it a case of polishing the existing roots or is it a completely new
> way of calculating them?

Polishing the existing roots would be a good quick fix, but what I am 
talking about is a general method for computing a sequence of roots of a 
special function in O(N) time. This  is a completely new way of calcuating 
them and could be used for zeros of plenty of other functions. Originally, 
it was developed for calculating roots for Gaussian quadrature rules so 
it applies to the various classical special functions used for quadrature.

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