Le samedi 12 avril 2008 à 21:51 -0700, Steven Watanabe a écrit :
The simplest implementation is to create a unique integer id for every new variable constructed. The format of a variable in the follwing discussion is (value, [(id1, alternate_value1), (id2, alternate_value2)]) where the alternate_values are the values that we would get if the value were increased by their uncertainties.
This approach to reliable computing is called "affine arithmetic". You may also want to look at other approaches called "Taylor arithmetic" (at first-order) or "slope arithmetic", as they are quite close. Obviously, since they are all first-order arithmetic, they all suffer from second-order terms. Yours is no exception. More precisely, what to do when one performs (x + ex) * (y + ey)? You decided to just discard ex*ey and hope it does not matter. There are two other solutions. First, create a new variable z that always corresponds to x*y. That way, you associate ex*ey to z and no information is lost. The drawback is that you will get a combinatorial explosion on the number of variables. Second, have a value that corresponds to all the higher-terms. The usual approach is to rely on an interval there. That way, you only have one additional term. But the drawback is that you can no longer detect dependencies on second-order terms, which tend to matter in iterative processes. The interval may be stored as lower/upper bounds, or as center/radius values, or as magnitude only (since the center is usually close to zero). An intermediate ground between these two solutions is to have second-order terms as new variables (so quadratic number only) while having an interval for dumping all the higher-order terms. A commonly-encountered variant is to add one single new term for each operation, so that first-order error compensations are properly handled. Anyway, notice that, when using an interval as a wastebin for terms, you can also use it to dump the rounding errors encountered during computations. That way, not only your error analysis is "reliable", but it is also guaranteed. Best regards, Guillaume