Re: Stieltjes polynomial

From: N A (testrope_at_[hidden])
Date: 2020-02-22 15:45:16

 But will both Stieltjes polynomials from the Legendre polynomials and Stieltjes polynomials with Legendre function of the second kind going to work as zeroes for the kronrod weights and nodes?
Because they both yield 0.53239 and 1.08169 for the same n=5 and x = 0.2 !

    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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