NIntegrate to compute LegendreP approximations to functions

*To*: mathgroup at smc.vnet.net*Subject*: [mg122953] NIntegrate to compute LegendreP approximations to functions*From*: "J. Jesús Rico Melgoza" <jerico at umich.mx>*Date*: Fri, 18 Nov 2011 06:23:15 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com

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=98ButtonBox[" ", 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

**Follow-Ups**:**Re: NIntegrate to compute LegendreP approximations to functions***From:*Bob Hanlon <hanlonr357@gmail.com>