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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: [mg71978] Re: How can I get this spiked Integral evaluated???
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Thu, 7 Dec 2006 06:25:20 -0500 (EST)
  • Organization: The Open University, Milton Keynes, UK
  • References: <el687j$2pv$1@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.
> 

Do you have any particular reason for not doing a symbolic integration 
(returning an exact result) first, then asking for a numeric result to 
any precision you might want?

In[1]:=
Integrate[Exp[-x^2], {x, -900, 1000}]

Out[1]=
1
- Sqrt[Pi] (Erf[900] + Erf[1000])
2

In[2]:=
N[%]

Out[2]=
1.77245

In[3]:=
N[%%, 30]

Out[3]=
1.77245385090551602729816748334

Regards,
Jean-Marc


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