Error propagation in adaptive Simpson algorithm

08012018, 02:23 PM
Post: #10




RE: Error propagation in adaptive Simpson algorithm
I finally have the time to learn about Adaptive Simpson's method.
I were misled to believe it is plain Simpson + correction (/15 only) ... Sorry It is not one integral with increasing steps for better accuracy, but increasing number of miniintegrals (2 intervals, thus 3 points) sum together. The question really is, when subdividing integral into a bunch of recursive miniintegrals, what should their tolerance be ? Miniintegrals don't talk to each other, so don't know which is the more important. Adaptive Simpson method have to be conservative, tolerance cut in half for each step. So, even if both splitted integrals equally important, total error still below tolerance. This is probably an overkill, but without knowing the integral, it is the best it can do. For one sided function, say exp(x), tolerance can really stay the same all the way down.  By setting a tolerance, adaptive Simpson rule ignore the "details", and concern itself with the dominant subdivided integrals, thus is faster. This is just my opinion, speedup by ignoring the details come with costs:
For example, this transformed exponential integral will not work well with Adatpive scheme: \(\int_0^{500}e^{x}dx \) = \(\int_{1}^{1}375(1u^2) e^{(125u (3  u^2) + 250)} du \) = 1  \(e^{500}\) ~ 1.0 OTTH, if used correctly, this is a great tool. For example: I(1000) = 4 \(\int_{0}^{\pi/2} cos(x) cos(2x) ... cos(1000x) dx \) = 0.000274258153608 ... Adaptive Simpson Method (eps = 1e9) were able to give 11 digits accuracy in 0.5 sec Romberg's Method can only reached 8 digits accuracy ... in 400 seconds ! Thanks, Claudio 

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