j0 (and y0) in the range 2 <= x < (p/2)*log(2)

[Steve Kargl]

On Thu, Sep 06, 2018 at 04:09:05AM +1000, Bruce Evans wrote:
> 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.

And as I noted, there is no documentation stating the approximations
pzero(x) and qzero(x) aren't approximations for the asymptotic series
P0(x) and Q0(x).  If you are correct, then pzero(x) and qzero(x) are
approximations to fudge*P0(x) and fudge*Q0(x).  What fudge is and how
it is determined is not documented.

-- 
Steve