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MathGroup Archive 2006

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Re: How can I get this spiked Integral evaluated???

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71985] Re: [mg71957] How can I get this spiked Integral evaluated???
  • From: Carl Woll <carlw at wolfram.com>
  • Date: Thu, 7 Dec 2006 06:25:30 -0500 (EST)
  • References: <200612061103.GAA02715@smc.vnet.net>

Zeno wrote:

>This is an integral with a spike that is not in the middle of the
>integral range. Here is the integral...
>
>NIntegrate[Exp[-x^2], {x, -900, 1000}]
>
>Mathemtaica 5.2.2 gives this error message....
>
> "Numerical \integration stopping due to loss of precision. Achieved
>neither the requested \PrecisionGoal nor AccuracyGoal; suspect one of
>the following: highly \oscillatory integrand or the true value of the
>integral is 0. If your \integrand is oscillatory on a (semi-)infinite
>interval try using the option \Method->Oscillatory in NIntegrate. 
>More?
>
>However, to use the option "Method->Oscillatory" one of the bounds of
>the integral must be infinity, so that would not work here. How do I
>get it to to the integral in the range -900 to 1000??
>The correct answer is 1.77245.
>  
>
One idea is stick the spike explicitly into the integration contour:

In[1]:=
NIntegrate[Exp[-x^2], {x, -900, 0, 1000}, MinRecursion -> 1]
Out[1]=
1.77245

Unfortunately, the MinRecursion option is necessary to get a correct 
result here.

Carl


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