Re: NIntegrate to compute LegendreP approximations to functions

• To: mathgroup at smc.vnet.net
• Subject: [mg123001] Re: NIntegrate to compute LegendreP approximations to functions
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Sun, 20 Nov 2011 05:35:46 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```Oops, disregard my earlier comment, as that's a DEFINITE integral.

Bobby

On Sat, 19 Nov 2011 14:19:04 -0600, DrMajorBob <btreat1 at austin.rr.com>
wrote:

> 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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