# Stieltjes polynomial

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## Stieltjes polynomial

Hi,

With regard to the article on Boost:
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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## Re: Stieltjes polynomial

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 <[hidden email]> wrote:

Hi,

With regard to the article on Boost:

## 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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[hidden email]
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## Re: Stieltjes polynomial

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 <[hidden email]> 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 <[hidden email]> wrote:

Hi,

With regard to the article on Boost:

## 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
[hidden email]
https://lists.boost.org/mailman/listinfo.cgi/boost-users

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## Re: Stieltjes polynomial

In reply to this post by Boost - Users mailing list
Hi,

For example the Legendre polynomials for degree n=5 and x = 0.2 is 0

On Friday, February 21, 2020, 06:54:27 PM GMT+4, Nick Thompson via Boost-users <[hidden email]> 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 <[hidden email]> wrote:

Hi,

With regard to the article on Boost:

## 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
[hidden email]
https://lists.boost.org/mailman/listinfo.cgi/boost-users

_______________________________________________
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[hidden email]
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## Re: Stieltjes polynomial

 In reply to this post by Boost - Users mailing list 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 <[hidden email]> 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 > <[hidden email]> 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 > [hidden email] > https://lists.boost.org/mailman/listinfo.cgi/boost-users> > _______________________________________________ > Boost-users mailing list > [hidden email] > https://lists.boost.org/mailman/listinfo.cgi/boost-users_______________________________________________ Boost-users mailing list [hidden email] https://lists.boost.org/mailman/listinfo.cgi/boost-users
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## Re: Stieltjes polynomial

 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.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 <[hidden email]> 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): 847-856John.>> Thanks>>>>> On Friday, February 21, 2020, 06:54:27 PM GMT+4, Nick Thompson via > Boost-users <[hidden email]> 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 > <[hidden email]> 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> [hidden email] > https://lists.boost.org/mailman/listinfo.cgi/boost-users>> _______________________________________________> Boost-users mailing list> [hidden email]> https://lists.boost.org/mailman/listinfo.cgi/boost-users_______________________________________________Boost-users mailing list[hidden email]https://lists.boost.org/mailman/listinfo.cgi/boost-users _______________________________________________ Boost-users mailing list [hidden email] https://lists.boost.org/mailman/listinfo.cgi/boost-users
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## Re: Stieltjes polynomial

In reply to this post by Boost - Users mailing list
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:

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 <[hidden email]> 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 <[hidden email]> 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 <[hidden email]> wrote:

Hi,

With regard to the article on Boost:

## 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 mailing list
[hidden email]
https://lists.boost.org/mailman/listinfo.cgi/boost-users
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## Re: Stieltjes polynomial

 In reply to this post by Boost - Users mailing list > 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 <[hidden email]> 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.Does this mean that we can generate different Stieltjes polynomials with different orthogonal polynomials and/or functions?Can you help me out please?ThanksOn Saturday, February 22, 2020, 01:26:11 PM GMT+4, John Maddock via Boost-users <[hidden email]> 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): 847-856John.>> Thanks>>>>> On Friday, February 21, 2020, 06:54:27 PM GMT+4, Nick Thompson via > Boost-users <[hidden email]> 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 > <[hidden email]> 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> [hidden email] >> _______________________________________________> Boost-users mailing list_______________________________________________Boost-users mailing list_______________________________________________ Boost-users mailing list [hidden email] https://lists.boost.org/mailman/listinfo.cgi/boost-users
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## Re: Stieltjes polynomial

In reply to this post by Boost - Users mailing list
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 <[hidden email]> 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:

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 <[hidden email]> 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 <[hidden email]> 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 <[hidden email]> wrote:

Hi,

With regard to the article on Boost:

## 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

_______________________________________________
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[hidden email]
https://lists.boost.org/mailman/listinfo.cgi/boost-users
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## Re: Stieltjes polynomial

 In reply to this post by Boost - Users mailing list 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 <[hidden email]> 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 <[hidden email]> 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.Does this mean that we can generate different Stieltjes polynomials with different orthogonal polynomials and/or functions?Can you help me out please?ThanksOn Saturday, February 22, 2020, 01:26:11 PM GMT+4, John Maddock via Boost-users <[hidden email]> 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): 847-856John.>> Thanks>>>>> On Friday, February 21, 2020, 06:54:27 PM GMT+4, Nick Thompson via > Boost-users <[hidden email]> 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 > <[hidden email]> 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> [hidden email] >> _______________________________________________> Boost-users mailing list_______________________________________________Boost-users mailing list _______________________________________________ Boost-users mailing list [hidden email] https://lists.boost.org/mailman/listinfo.cgi/boost-users