Re: Stieltjes polynomial

From: Nick Thompson (nathompson7_at_[hidden])
Date: 2020-02-22 14:07:39

I get the feeling that you want to compute the coefficients of the polynomial in the standard basis:

c0 + c1*x + ...

Unfortunately, this is a bad idea, because the computation is horrifically ill-conditioned. That's why the boost version expands the Legendre-Stieltjes polynomials in the Legendre polynomial basis-this is well-conditioned. I vaguely recall that expansion in the Chebyshev basis is also well-conditioned, but we succeeded in the Legendre basis and were happy.

The code, to my eyes, is legible, with references to papers and equations within papers:

https://github.com/boostorg/math/blob/develop/include/boost/math/special_functions/legendre_stieltjes.hpp

What are you trying to accomplish by computing these polynomials? The only application I know of is Gauss-Kronrod quadrature, so I'd be interested if you have another application . . .

‐‐‐‐‐‐‐ Original Message ‐‐‐‐‐‐‐
On Friday, February 21, 2020 10:25 PM, N A <testrope_at_[hidden]> wrote:

> Hi
>
> The Legendre polynomials (Lp) of degree n=5 and x=0.2 is 0.30752 and according to Boost article, the Legendre-Stieltjes polynomials (LSp) of degree n=5 and x=0.2 is 0.53239.
>
> So if I want to compute the LSp for n=6, how do I do it? What is the formula you are using to be able to calculate the LSp for any nth degree?
>
> If a recurrence relation is not possible, then is there a closed form mathematical representation to calculate any nth degree LSp?
>
> Thanks
>
> On Friday, February 21, 2020, 06:54:27 PM GMT+4, Nick Thompson via Boost-users <boost-users_at_[hidden]> wrote:
>
> What precisely are you trying to compute? Are you trying to find the coefficients of the polynomials in the standard basis? Are you trying to evaluate them at a point?
>
> Note that the Legendre-Stieltjes polynomials do not satisfy three-term recurrence relations, and so recursive rules (depending on what precisely you mean by that) are not available.
>
>    Nick
>
> ‐‐‐‐‐‐‐ Original Message ‐‐‐‐‐‐‐
> On Wednesday, February 19, 2020 12:07 PM, N A via Boost-users <boost-users_at_[hidden]> wrote:
>
>> Hi,
>>
>> With regard to the article on Boost:
>> [Legendre-Stieltjes Polynomials - 1.66.0](https://www.boost.org/doc/libs/1_66_0/libs/math/doc/html/math_toolkit/sf_poly/legendre_stieltjes.html)
>>
>> Legendre-Stieltjes Polynomials - 1.66.0
>>
>> Can anyone help me to compute the stieltjes polynomials please? I'm coding in VBA and I'm looking for some recursive rules to calculate same.
>>
>> Thanks
>> Vick
>
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