Date: Thu, 6 Sep 2018 04:09:05 +1000 (EST) From: Bruce Evans <brde@optusnet.com.au> To: Steve Kargl <sgk@troutmask.apl.washington.edu> Cc: freebsd-numerics@freebsd.org Subject: Re: j0 (and y0) in the range 2 <= x < (p/2)*log(2) Message-ID: <20180906034525.A2959@besplex.bde.org> In-Reply-To: <20180905152104.GA26453@troutmask.apl.washington.edu> References: <20180903235724.GA95333@troutmask.apl.washington.edu> <20180905201540.D1142@besplex.bde.org> <20180905152104.GA26453@troutmask.apl.washington.edu>
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On Wed, 5 Sep 2018, Steve Kargl wrote: > On Wed, Sep 05, 2018 at 10:06:29PM +1000, Bruce Evans wrote: >> On Mon, 3 Sep 2018, Steve Kargl wrote: >> >>> Anyone know where the approximations for j0 (and y0) come from? >> >> I think they are ordinary minimax rational approximations for related >> functions. As you noticed, the asymptotic expansion doesn't work below >> about x = 8 (it is off by about 10% for j0(2). But we want to use the >> single formula given by the asymptotic expansion for all the subintervals: > > I've scoured the literature and web for methods of computing > Bessel functions. These functions are important to my real > work. I have not found any paper, webpage, documentation, etc. > that describes what "the related functions" are. They are just the functions in the asymptotic expansion with errors corrected as I discussed. >> ... >> However, we use the asympotic formula for all cases above x = 2. It is >> off by at least 10% unless pzero and qzero are adjusted. But 10% is not >> much, and adjusting only pzero and qzero apparently works to compensate >> for this. To use the same formula for all cases, the complicated factors >> cc, ss and sqrt(x) must not be folded into pzero and qzero; instead these >> factors supply some of the errors which must be compensated for. >> >> To find the adjustments, I would try applying the same scale factor to >> the nominal pzero and qzero. E.g., for j0, the error factor is >> invsqrtpi*(u*cc-v*ss)/sqrt(x) / j0(x). Divide the nominal pzero and qzero >> by this. Apply normal minimax methods to the modfided pzero and qzero. > > Obviously, I had not thought about scaling P0(x) and Q0(x) with a > common factor to force agreement between j0(x) and its approximation. I think I understand why we use the asymptotic formula as a base. If we choose a minimax function for each interval, this would need to have a very high order. Much the same order as the rational function part of the asymptotic expansion. But the latter has a clever grouping of terms using cos and sin and in the internals of P0 and Q0. This makes it faster to calculate and also probably has better accuracy. Brucehome | help
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