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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




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