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From: Jan Simon on 26 May 2010 05:55
Dear James, dear OP!

> Yes, I agree. This discussion so far seems to be missing one important thing, the accuracy of the original data. Take this simple list:
>
> -1 1e-20 1
>
> The obvious "best" sum is 1e-20, right?

You hit the point! 1e-20 is the mathematically correct answer. But usually mathematics with floating point numbers are used on a computer to model physical measurements. A well implemented model has to consider the measurement errors, so actually this sum must be calculated as e.g. (if the values have been measured with a relative error of 1e-16 - unlikely for real world problems...):
{-1+-1e15} + {1e-20+-1e-35} + {1+-1e-15}
Here the curly braces means a variation of the measurement values. The result would be an interval!
E.g. for the numerical integration of a function, the sensitivities to the initial values and parameters have to be calculated also - e.g. by a sufficiently small variation of the inputs- otherwise the resulting number is more or less meaningless.
A trustworthy comparison of numerical algorithms means a check, if the resulting intervals overlap considering the accuracy (or precision?!) of the input data.

INTLAB written by Rump is a very well implemented interval library written in Matlab:
http://www.ti3.tu-harburg.de/~rump/intlab/

Kind regards, Jan
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