Re: Stieltjes polynomial

From: N A (testrope_at_[hidden])
Date: 2020-02-24 06:25:20

 I've been able to go through the paper and I have potentially succeeded in calculating the coefficients by means of Gaussian elimination. But I want to make sure, I got it right!
So can you please check for me x=0.2 and n=6,7,8?
Thanks a lot!Vick

    On Saturday, February 22, 2020, 06:11:33 PM GMT+4, Nick Thompson <nathompson7_at_[hidden]> wrote:  
 
 > 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.
>
> Â Â  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>
>>
>>
>> Â Â Â  
>>
>>
>>Â  Â  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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