Numerical integration: Casio fx115ES, TI82, HP42S, and HP32SII Message #28 Posted by Karl Schneider on 4 Dec 2009, 3:03 a.m., in response to message #26 by Mike Morrow
Quote:
The evidence shows that fx991ES/115ES quadrature is blazingly fast (more than two orders of magnitude for this example!) compared to that of any HP handheld.
I don't know much about the GaussKronrod quadrature method, but it would be a most welcome replacement (or better, a selectable option) for the methodolgy traditionally implemented for the last third of a century on HP calculators.
Ya learn something every day! I was puzzled about what you were doing in this thread, because I have the predecessor fx115MS, which looks almost identical. It uses Simpson's Rule for integration, just like the 1981 Casio fx3600P of mine.
A online review for the fx115ES confirms that the new model upgraded to GaussKronrod. Maybe there was more to the fx115 replacement than met the eye.
Upscale TI models have used GaussKronrod for many years. My 1993 TI82 generally outperforms my HP models on integration, other than that the user must ensure that the integrand function is defined at the endpoints:
fnInt(sqrt(X)/(X1)1/ln(X),X,1E12,11E12,1E12)
.036489974 (displayed in 54 seconds)
0.036489973978679 (revealed)
Quote:
I tried to get a 10digit result for integration of the original function f(x)=(sqrt(x)/(x1)1/ln(x)) (for 0 <= x <= 1) on my HP42S with ACC set to 10^10. After more than two hours of run time, I gave up.
It should be noted that the ACC parameter on the HP42S is not the same as FIX on the "lesser" RPN models (prior to the HP33s)  or even the accuracy factor on nonHP calcuators using GaussKronrod. ACC is a multiplicative (not absolute) uncertainty factor applied to the integrand function. With the value of the function f(x) well below 0.1 for 0 < x < 1, the uncertainty of the function at almost all points for ACC = 1E10 is well below 1E11. This tight tolerance drives a large number of function evalulations, in order to achieve calculated estimates of the integral that do not change by more than the small totalintegral uncertainty as evermore evaluations are taken.
On my  er, perhaps loathsome  HP32SII, I got 0.0364899740909 in 605 seconds with a FIX 9 setting. The deviation from the correct answer of 0.036489973978576 is 1.123E10  well within the maximum error of 0.5 * 1E9 * (10) = 5E10 estimated for FIX 9.
In order to get a corresponding answer on the HP42S, I set ACC to a value such that the mean value of the function times ACC approximately equaled the FIX 9 uncertainty: ACC = 5.00E10 / 0.0365 = 1.37E08. This yielded an integral of 0.036489974091, with an estimated error of 4.99963E10, in about 1000 seconds.
Please see my sole contribution to the HP Articles Forum (#556) for more information.
 KS
Edited: 5 Dec 2009, 11:54 p.m. after one or more responses were posted
