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Re: NIntegrate to compute LegendreP approximations to functions


The constant term in an indefinite integral is arbitrary, so what do you 
mean by computing it "properly"?

This may be the plot you want:

Plot[{u, -1 + Sum[c[k] LegendreP[k, t], {k, 0, 20}]}, {t, -1, 1}]

Bobby

On Sat, 19 Nov 2011 05:46:10 -0600, J. Jes=FAs Rico Melgoza 
<jerico at umich.mx> wrote:

> 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
>>
>
>


--
DrMajorBob at yahoo.com



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