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Re: NIntegrate to compute LegendreP approximations to functions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg122962] Re: NIntegrate to compute LegendreP approximations to functions
  • From: Bob Hanlon <hanlonr357 at gmail.com>
  • Date: Fri, 18 Nov 2011 07:50:25 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <201111181123.GAA06494@smc.vnet.net>

Do the integration once.

u = Sign[t];

c[k_] = Simplify[
  (2 k + 1)/2 Integrate[u LegendreP[k, t], {t, -1, 1}],
  Element[k, Integers]]

((1 + 2*k)*Sqrt[Pi])/(2*Gamma[1 - k/2]*Gamma[(3 + k)/2])


Bob Hanlon


2011/11/18 "J. Jes=FAs Rico Melgoza" <jerico at umich.mx>:
>
> Hello
> I am approximating general scalar functions via orthogonal series. I am
> using LegendreP polynomials.
> As an example, I have approximated a Sign function. The coefficients
> have been calculated as follows:
>
> n = 20;
> u = Sign[t];
> N[Table[(2 k + 1)/2 Integrate[u LegendreP[k, t], {t, -1, 1}], {k, 0,
> n}]]
>
> Everything works well but I would like to speed up computations since
> for large values of n, Integrate takes long computations times. I need
> to speed up the process since in general I will be approximating
> multi-variable functions. I have tried NIntegrate but I get multiple
> messages such as
>
> NIntegrate::slwcon :  "Numerical integration converging too slowly;
> suspect \
> one of the following: singularity, value of the integration is 0, highly
> \
> oscillatory integrand, or WorkingPrecision too small. =91=99=98Butt=
onBox["
> ",
> Appearance->{Automatic, None},
> BaseStyle->"Link",
> ButtonData:>"paclet:ref/message/NIntegrate/slwcon",
> ButtonNote->"NIntegrate::slwcon"]"
>
> NIntegrate is a very complete function in Mathematica, so much that it
> has been rather difficult to find an adequate combination of  a method
> and a strategy of integration that would improve the timing of
> Integrate.
>
> Could anyone give me some advice?
>
> Jesus Rico-Melgoza



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