Re: NIntegrate to compute LegendreP approximations to functions

*To*: mathgroup at smc.vnet.net*Subject*: [mg122979] Re: NIntegrate to compute LegendreP approximations to functions*From*: "J. Jesús Rico Melgoza" <jerico at umich.mx>*Date*: Sat, 19 Nov 2011 06:46:10 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201111181123.GAA06494@smc.vnet.net> <201111181250.HAA07958@smc.vnet.net>

Thanks for the advise. Though, I don't see why the constant term is not calculated properly. The resulting approximation in Plot[{u, Sum[c[k] LegendreP[k, t], {k, 0, 20}]}, {t, -1, 1}] has a different c[0]. J. Rico El 18/11/2011, a las 06:50, Bob Hanlon escribi=F3: > 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=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 >

**References**:**NIntegrate to compute LegendreP approximations to functions***From:*"J. Jesús Rico Melgoza" <jerico@umich.mx>

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