From owner-freebsd-numerics@freebsd.org  Tue Sep  4 04:10:12 2018
Return-Path: <owner-freebsd-numerics@freebsd.org>
Delivered-To: freebsd-numerics@mailman.ysv.freebsd.org
Received: from mx1.freebsd.org (mx1.freebsd.org [IPv6:2610:1c1:1:606c::19:1])
 by mailman.ysv.freebsd.org (Postfix) with ESMTP id E067DFE00B9
 for <freebsd-numerics@mailman.ysv.freebsd.org>;
 Tue,  4 Sep 2018 04:10:12 +0000 (UTC)
 (envelope-from sgk@troutmask.apl.washington.edu)
Received: from troutmask.apl.washington.edu (troutmask.apl.washington.edu
 [128.95.76.21])
 (using TLSv1.2 with cipher ECDHE-RSA-AES256-GCM-SHA384 (256/256 bits))
 (Client CN "troutmask", Issuer "troutmask" (not verified))
 by mx1.freebsd.org (Postfix) with ESMTPS id 6ACB5850DD
 for <freebsd-numerics@freebsd.org>; Tue,  4 Sep 2018 04:10:12 +0000 (UTC)
 (envelope-from sgk@troutmask.apl.washington.edu)
Received: from troutmask.apl.washington.edu (localhost [127.0.0.1])
 by troutmask.apl.washington.edu (8.15.2/8.15.2) with ESMTPS id w844AAAG096219
 (version=TLSv1.2 cipher=DHE-RSA-AES256-GCM-SHA384 bits=256 verify=NO);
 Mon, 3 Sep 2018 21:10:10 -0700 (PDT)
 (envelope-from sgk@troutmask.apl.washington.edu)
Received: (from sgk@localhost)
 by troutmask.apl.washington.edu (8.15.2/8.15.2/Submit) id w844AAbc096218;
 Mon, 3 Sep 2018 21:10:10 -0700 (PDT) (envelope-from sgk)
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>
Reply-To: sgk@troutmask.apl.washington.edu
References: <20180903235724.GA95333@troutmask.apl.washington.edu>
 <ae55265d8d544cdc8a3ebc4314e1d01c@missouri.edu>
MIME-Version: 1.0
Content-Type: text/plain; charset=us-ascii
Content-Disposition: inline
In-Reply-To: <ae55265d8d544cdc8a3ebc4314e1d01c@missouri.edu>
User-Agent: Mutt/1.10.1 (2018-07-13)
X-BeenThere: freebsd-numerics@freebsd.org
X-Mailman-Version: 2.1.27
Precedence: list
List-Id: "Discussions of high quality implementation of libm functions."
 <freebsd-numerics.freebsd.org>
List-Unsubscribe: <https://lists.freebsd.org/mailman/options/freebsd-numerics>, 
 <mailto:freebsd-numerics-request@freebsd.org?subject=unsubscribe>
List-Archive: <http://lists.freebsd.org/pipermail/freebsd-numerics/>
List-Post: <mailto:freebsd-numerics@freebsd.org>
List-Help: <mailto:freebsd-numerics-request@freebsd.org?subject=help>
List-Subscribe: <https://lists.freebsd.org/mailman/listinfo/freebsd-numerics>, 
 <mailto:freebsd-numerics-request@freebsd.org?subject=subscribe>
X-List-Received-Date: Tue, 04 Sep 2018 04:10:13 -0000

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