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Subject: Re: Guaranteed precision at numerical integration?
Date: Wed, 2 Nov 2011 23:37:26 +0000 (UTC)
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"John D'Errico" <woodchips@rochester.rr.com> wrote in message <j8shit$ib4$1@newscl01ah.mathworks.com>...
> "Triantafyllos" wrote in message <j8s49o$4hf$1@newscl01ah.mathworks.com>...
> > Hello,
> > 
> > in order to prove a mathematical statement, I need check the value of an integral which is evaluated numerically in Matlab.
> > 
> > Now, which guarantee do I have that the value is correct up to say the 5th significant digit? Since, if it is only correct some 99.9% of the time, I apparantly can not use this value to rigorously prove the mathematical statement..
> > 
> > Infos/Links on this topic are highly appreciated! Thank you very much in advance,
> > Best regards.
> 
> You CANNOT guarantee any fixed tolerance on a numerical
> integration. Period.
> 
> One can always provide a function that will cause any adaptive
> integration tool such as quad to fail any tolerance you choose.
> 
> Period.
> 
> John

I thought about if there was an algorithm which only stopped after checking via the standard error terms (using derivatives) that the desired precision has been reached? However, I understand that evaluating these error terms usually involves evaluating derivatives in certain ranges, something which may in most cases cause at least similar problems as the original problem.

And according to your statement John, there is no such algorithm for general functions, which sounds reasonable.