Re: NIntegrate to compute LegendreP approximations to functions

• To: mathgroup at smc.vnet.net
• Subject: [mg122999] Re: NIntegrate to compute LegendreP approximations to functions
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Sun, 20 Nov 2011 05:35:24 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```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},
>>> 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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