Accuracy of Integral with epsilon

10092021, 08:37 PM
Post: #18




RE: Accuracy of Integral with epsilon
(10092021 02:27 AM)Paul Dale Wrote:(10092021 02:22 AM)Valentin Albillo Wrote: Are you saying that a complex stateoftheart TanhSinh method requires 27 evaluations of the integrand (the constant 2) to integrate the constant 2 from 10 to 10 ? Really !? There is never a guarantee that a given function is constant by probing a few points. That should be obvious, no? Naive implementations of GaussLegendre quadrature methods may stop at two or three points to fit a constant (polynomial). However, it would be detrimental to give up that early. Evidently, calculators run quite a bit longer on this function for a good reason. My point about TanhSinh was simply that this method doesn't require a huge number of points to converge accurately and it is used in the WP 34S... so make your own conclusions. TanhSinh simply works well for this type of function based on its syntactical structure (duh). More importantly, the function is NOT numerically constant in its present nonsimplified form, which is to be expected: noise increases quickly towards 10 and beyond. This noise is already in the order of 10^2 at x=18. Tabulating the error in Excel from 10 to 8.9 gives: 5.31248E08 6.01538E08 5.39791E08 1.77564E08 1.86315E08 1.64802E08 5.18465E10 8.54069E09 1.47008E08 1.40072E08 6.26628E09 5.77673E09 This amount of noise is sufficient to trip up quadrature methods. Of course, the noise may differ with nonIEEE 754 double precision and different implementations of the constituent functions EXP, SINH and COSH. However, a general consequence of the noise and floating point limitations, you can either get lucky to get to the exact result 40 with a few points or unlucky, which can cost you a great deal of time wasted to evaluate points in the presence of noise. The WP 34S and qthsh points reported show exactly what I mean. Also, attempting to push the accuracy beyond 10^8 (or about) is pointless. There is too much noise to make definitive conclusions.  Rob "I count on old friends"  HP 71B,PrimeTi VOY200,Nspire CXII CASCasio fxCG50...Sharp PCG850,E500,2500,1500,14xx,13xx,12xx... 

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