Numerical integration methods

07252022, 10:36 PM
Post: #21




RE: Numerical integration methods
That is what is going on! When, on the Classpad 400, I write integral from 1 to 100 of x^(2(e^((1/x)^3)1)), I get the 99.5996+ answer!


07262022, 05:18 PM
Post: #22




RE: Numerical integration methods
I never considered that as a possible interpretation of the original expression. Well done for figuring out what was going on! I'm glad that's sorted.
Nigel (UK) 

07312022, 04:59 PM
Post: #23




RE: Numerical integration methods
(07242022 08:55 PM)Wes Loewer Wrote: (Note: It's possible that I am mistranslating "ordre". What the paper called "ordre 30" seems to be "degree 29" (ie, 30 terms), but I could be wrong about this. Francophones, please correct me if I am mistaken.)"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... Quote:Since the 15node GaussKronrod is exact up to degree 22 and the Prime's method is exact up to degree 29, it would seem that the Prime's method is a bit better with only a small added overhead. A slightly more accurate calculation per iteration could occasionally reduce the need for further recursion. It would be interesting to do some comparisons of these two methods with some wellbehaved functions as well as more "temperamental" ones.Yes, it is important not to reevaluate 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(i30i14) is in h^14 and err2=abs(i30i6) in h^6 

08012022, 03:43 AM
Post: #24




RE: Numerical integration methods
(07312022 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. (07312022 04:59 PM)parisse Wrote: Yes, it is important not to reevaluate the function for efficiency reasons. The idea is to estimate the error of the order 30 method in h^30 by 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. 

08012022, 03:01 PM
Post: #25




RE: Numerical integration methods
(08012022 03:43 AM)Wes Loewer Wrote: 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. This is my guess ... err1 = k1 * h^15 err2 = k2 * h^7 integral error estimate = err1*(err1/err2)^2 = k1*(k1/k2)^2 * h^(15+8*2) = k * h^31 If (k, k1, k2) are similar in size, constant term matched too. Exponents cannot be picked in random; it had to produce O(h^31) on the right. If we use 5 or 7 nodes, instead of square, it would need some noninteger exponents. 

08012022, 03:46 PM
Post: #26




RE: Numerical integration methods
(08012022 03:01 PM)Albert Chan Wrote: This is my guess ... Okay, I must be missing something. Why does it have to produce O(h^31) on the right? What if the algorithm turned out to be O(h^33) or some other order? How would using a different number of nodes produce noninteger exponents? Even if it did produce noninteger exponents, what would be wrong with that? Sorry if I'm missing something obvious. 

08012022, 07:01 PM
Post: #27




RE: Numerical integration methods
The exponent of the error estimate must match the error majoration of the best quadrature.


08012022, 07:24 PM
Post: #28




RE: Numerical integration methods  
08042022, 12:54 PM
Post: #29




RE: Numerical integration methods
(07192022 05:16 PM)parisse Wrote: int uses an adaptive method, as described here in French (Hairer: https://www.unige.ch/~hairer/poly/chap1.pdf). Bernard, out of interest, is a version of the Risch Algorthm still used for symbolic integration in XCAS/HP Prime? 

08052022, 09:45 AM
Post: #30




RE: Numerical integration methods
Yes, the rational version is implemented.


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