Re: About HP33S integration Message #4 Posted by Karl Schneider on 26 July 2006, 3:06 a.m., in response to message #3 by Antonio Maschio (Italy)
Antonio 
The user's manuals for the HP15C and the HP34C explain the philosophy and implementation of numerical integration on those models. This formed the basis of the approach used on subsequent RPN models, for which the numerical integration function is not nearly as well documented.
I'm not sure how accessible (or even available) the manuals for the HP15C and the HP34C are, in Italian.
The idea is this: It is not reasonable for a user to directly specify a required maximum error for the result of a numerical integration, because the exact answer is not known. Instead, the user specifies the accuracy (or tolerance) of the integrand function by using the displaymode settings (FIX, SCI, or ENG). The calculator produces a "worstcase" estimated error by integrating the tolerance of the integrand function over the range of integration. Assuming that the integral is calculated accurately, the estimated error will likely exceed the actual error by a significant percentage amount.
Original method of estimating error of integration:
This original method is implemented as described on the HP34C, HP15C, HP32S, HP32SII, and HP41 Advantage Pac. (On the HP42S, the uncertainly is entered as an "accuracy factor" multiplier variable, without using FIX/SCI/ENG.)
If FIX 2 is set, the displayed value of the integrand function will be within +/ 0.005 of the full numerical value, so the "actual" value (whatever it is) will be assumed to be within the same tolerance. This absolute uncertainty can be visualized as a ribbon or band of fixed width. Thus, the estimated maximum error will be 0.005*(ba), where a and b are the limits of integration.
SCI and ENG are a bit trickier. If SCI 2 is set, the displayed value of the integrand function will be within 0.005 x 10^{n} of the full numerical value, where n is the base10 exponent displayed. This is a stepwise relative uncertainty. The estimated error must be calculated by integration, and the tolerance "band" widens by an order of magnitude when the integrand function increases beyond an integer power of 10.
HP33S estimated error of integration:
Perhaps the different error calculations on the HP33S are "simpler" and more consistent between FIX, SCI, and ENG, but I believe that the user is being poorly served by the unnecessarilycoarse estimates of error. It is quite clear that the HP33S calculated integral of f(x) = x^{2} between x = 0 and x = 4 is much closer to the actual answer of 21 1/3 than the displayed error of 0.21 would indicate.
If the HP33S is estimating integration error using FIX N, SCI N, ENG N as follows:
Estimated_error = Calculated_integral * 10^{N}
Well... that's just not very good.
 KS
Edited: 26 July 2006, 3:14 a.m.
