j0 (and y0) in the range 2 <= x < (p/2)*log(2)
Bruce Evans
brde at optusnet.com.au
Wed Sep 5 18:09:17 UTC 2018
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.
Bruce
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