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Re: Precision issues


mickey wrote:
> Hi,
> 
> I am calculating certain integrals numerically and get back a number. 
> Now, is it possible to determine how many digits is that answer accurate 
> to?
> 
> E.g.,
> 
> NIntegrate[ Exp[-p^2 - q^2], {p, 0, 10}, {q, 0, 10}, Method ->
>      MonteCarlo[24], MaxPoints -> 1000000]
> 
> Gives,
> 
> 0.791249
> 
> How many digits is this answer accurate to?

You are using machine-precision numbers; therefore you cannot know 
since, "Machine numbers, [...], always contain the same number of 
digits, and maintain no information on their precision. [1]"

In[1]:=
sol = NIntegrate[Exp[-p^2 - q^2], {p, 0, 10},
    {q, 0, 10}, Method -> MonteCarlo[24],
    MaxPoints -> 1000000]

Out[1]=
0.7912492023745136

In[2]:=
Precision[sol]

Out[2]=
MachinePrecision

If you desire to be guarantee a specific level of precision, or at least 
to know when Mahtematica failed to reach the desired precision, you must 
use arbitrary-precision numbers. See the options PrcecisionGoal and 
WorkingPrecision on the online help.

Regards,
Jean-Marc

[1] " The Mathematica Book Online  /  Advanced Mathematics in 
Mathematica  /  Numbers  /  3.1.4 Numerical Precision" 
http://documents.wolfram.com/mathematica/book/section-3.1.4


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