Re: Stieltjes polynomial

From: N A (testrope_at_[hidden])
Date: 2020-02-22 15:40:32

 Yes, the purpose is to code the Gauss-Kronrod quadrature. Thanks for the link, but I'm not familiar with hpp.
Can you help me out please with the code? I'm ok with Python and VBA!
Or you can tell me the math equation at each step, whichever is more convenient for you.
Thanks

    On Saturday, February 22, 2020, 06:07:46 PM GMT+4, Nick Thompson <nathompson7_at_[hidden]> wrote:  
 
 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:Â 
Legendre-Stieltjes Polynomials - 1.66.0

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Legendre-Stieltjes Polynomials - 1.66.0

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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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