Numerical integration methods
08-01-2022, 03:43 AM
Post: #24
 Wes Loewer Senior Member Posts: 453 Joined: Jan 2014
RE: Numerical integration methods
(07-31-2022 04:59 PM)parisse Wrote:  "order" may mean max degree of polynomial where the method is exact, or the exponent in the step of the error, up to the author choice...

I see. The order of the error, not the order of the polynomial, in this case. Thank you.

(07-31-2022 04:59 PM)parisse Wrote:  Yes, it is important not to re-evaluate the function for efficiency reasons. The idea is to estimate the error of the order 30 method in h^30 by
err1*(err1/err2)^2
where err1=abs(i30-i14) is in h^14 and err2=abs(i30-i6) in h^6

This method works beautifully, but I am fuzzy about some of the details.

For instance, why *(err1/err2)^2 ? Why not *(err1/err2) or *(err1/err2)^3 ? Was ^2 derived mathematically, or experimentally? I'm guessing it was experimentally determined to give a reasonable error approximation.

And why does the 3rd calculation use 6 nodes? Why not 5 or 7? I realize you want the 3rd calculation to be different enough (but not too different) from the 2nd calculation to determine the error. Was 6 determined experimentally to produce an optimal error estimate? Or is there something mathematical that makes 6 the best choice? Using 6 leaves the entire middle third of the interval unevaluated. Seems like including the center node would have been advantageous.
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