Re: Stieltjes polynomial

From: Nick Thompson (nathompson7_at_[hidden])
Date: 2020-02-22 14:11:27

> Does this mean that we can generate different Stieltjes polynomials with different orthogonal polynomials and/or functions?

Yes, you can expand every polynomial in every other complete polynomial basis. The basis you select should make the conversion from the original basis well-conditioned.

‐‐‐‐‐‐‐ Original Message ‐‐‐‐‐‐‐
On Saturday, February 22, 2020 6:26 AM, N A via Boost-users <boost-users_at_[hidden]> wrote:

> What is the "triangular system of equations" that need to be solved? And how to solve it?
>
> I'm not familiar with these terms!
>
> However, I came across another article beside yours that dealt with Stieltjes polynomials. Yours deal with Legendre polynomials-Stieltjes polynomials, but theirs deal with Legendre function of the second kind with regard to Stieltjes polynomials.
>
> They have a mathematica code, which I don't quite understand but their code yields 1.08169 for the same n and x as below.
>
> https://tpfto.wordpress.com/2019/04/14/stieltjes-polynomials-and-gauss-kronrod-quadrature/
>
> Does this mean that we can generate different Stieltjes polynomials with different orthogonal polynomials and/or functions?
>
> Can you help me out please?
> Thanks
>
> On Saturday, February 22, 2020, 01:26:11 PM GMT+4, John Maddock via Boost-users <boost-users_at_[hidden]> wrote:
>
> On 22/02/2020 03:25, N A via Boost-users 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?
>
> Please see Patterson, TNL. "The optimum addition of points to quadrature
> formulae." Mathematics of Computation 22.104 (1968): 847-856
>
> John.
>
>>
>> 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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