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 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 | | | | 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 _______________________________________________ Boost-users mailing list Boost-users@lists.boost.org https://lists.boost.org/mailman/listinfo.cgi/boost-users

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

_______________________________________________ Boost-users mailing list Boost-users@lists.boost.org mailto:Boost-users@lists.boost.org https://lists.boost.org/mailman/listinfo.cgi/boost-users

_______________________________________________ Boost-users mailing list Boost-users@lists.boost.org https://lists.boost.org/mailman/listinfo.cgi/boost-users

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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 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

_______________________________________________ Boost-users mailing list Boost-users@lists.boost.org mailto:Boost-users@lists.boost.org https://lists.boost.org/mailman/listinfo.cgi/boost-users

_______________________________________________ Boost-users mailing list Boost-users@lists.boost.org https://lists.boost.org/mailman/listinfo.cgi/boost-users

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Hi, I haven't heard from you. I hope all is well? When you get the time, please check x=0.2 and n=6,7 & 8 for me. Thanks On Monday, February 24, 2020, 10:25:20 AM GMT+4, N A wrote: 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

_______________________________________________ Boost-users mailing list Boost-users@lists.boost.org mailto:Boost-users@lists.boost.org https://lists.boost.org/mailman/listinfo.cgi/boost-users

_______________________________________________ Boost-users mailing list Boost-users@lists.boost.org https://lists.boost.org/mailman/listinfo.cgi/boost-users

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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 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_fun... 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 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 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 | | | | 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 _______________________________________________ Boost-users mailing list Boost-users@lists.boost.org https://lists.boost.org/mailman/listinfo.cgi/boost-users