Date: Mon, 3 Sep 2018 21:10:10 -0700 From: Steve Kargl <sgk@troutmask.apl.washington.edu> To: "Montgomery-Smith, Stephen" <stephen@missouri.edu> Cc: "freebsd-numerics@freebsd.org" <freebsd-numerics@freebsd.org> Subject: Re: j0 (and y0) in the range 2 <= x < (p/2)*log(2) Message-ID: <20180904041010.GA96191@troutmask.apl.washington.edu> In-Reply-To: <ae55265d8d544cdc8a3ebc4314e1d01c@missouri.edu> References: <20180903235724.GA95333@troutmask.apl.washington.edu> <ae55265d8d544cdc8a3ebc4314e1d01c@missouri.edu>
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On Tue, Sep 04, 2018 at 03:56:28AM +0000, Montgomery-Smith, Stephen wrote: > A quick google search turned up this > > https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf > > which has the functions p0 and q0. Maybe this was the basis of this code. I've read that paper. It uses |x| > 45 for the cut over to the large argument asymptotic expansion. One of the primary results for that paper is the development of new approximations that are robust near zeros of Jn(x). In the the discussion of the results, the paper notes the use of a double-double representation for intermediate results. A&S claims that the remainder in truncating the series does not exceed the magnitude of the first neglected term. If you set x = 2 and compute the terms in p0(x), one finds the smallest term is about |pk| = 1e-4. -- Steve
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