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Re: Precision issues
*To*: mathgroup at smc.vnet.net
*Subject*: [mg73543] Re: Precision issues
*From*: "Robert Dodier" <robert.dodier at gmail.com>
*Date*: Wed, 21 Feb 2007 01:53:40 -0500 (EST)
*References*: <erelv2$7ol$1@smc.vnet.net>
mickey wrote:
> 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?
For MC (and quasi-MC, a somewhat different method) the approximate
error is bounded by V(f) times D*(x_1, ..., x_n), where V(f) is the
variation (in a specific technical sense) of the integrand f, and
D* is the so-called star discrepancy of the points on which the
integrand is evaluated. It is remarkable that the error can be
decomposed into two terms such that one depends only on the
integrand and the other depends only on the method.
For the MC method, the expected value of the star discrepancy is
sqrt((log log n)/(2 n)). If you could estimate V(f), you would be able
to put an approximate bound on the error of approximation.
Instead of attempting to bound the error, you could ask what is the
variance of the result. For the MC method the variance is proportional
to 1/n, so the standard deviation is proportional to 1/sqrt(n).
Essentially you have to take 100 times as many samples to get 1 more
decimal place in the result.
These articles
http://en.wikipedia.org/wiki/Monte_Carlo_integration
http://en.wikipedia.org/wiki/Quasi-Monte_Carlo_method
http://en.wikipedia.org/wiki/Low-discrepancy_sequence
have more about these topics.
HTH
Robert Dodier
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