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

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

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