Re: [Boost-users] Stieltjes polynomial

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 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 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-kronr... 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 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 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 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_pol...

   

    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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