HP HP-32S - Conditions that Could Cause Incorrect Results

As explained in chapter 8, the uncertainty of the final approximation
is a number derived from the display format, which specifies the un
certainty for the function. Attheendof each iteration, the algorithm
compares the approximation calculated during that iteration with the
approximations calculated during two previous iterations. If the dif
ference between any of these three approximations and the othertwo
is less than the uncertainty tolerable in the final approximation, the
calculations ends, leaving the current approximation in the X-register
and its uncertainty in the Y-register.
Itis extremely unlikely that the errors in each of three successive ap
proximations—that is, the differences between the actual integral and
the approximations—would all be larger than the disparity among the
approximations themselves. Consequently, the error in the final ap
proximation will be less than its uncertainty (provided that f(x) does
not vary rapidly). Although we can't know the error in the final ap
proximation, the error is extremely unlikely to exceed the displayed
uncertainty of the approximation. In other words, the uncertainty esti
mate in the Y-register is an almost certain "upper bound* on the
difference between the approximation and the actual integral.
Conditions That Could Cause Incorrect
Results
Although the integration algorithm in the HP-32S is one of the best
available, in certain situations it—like all other algorithms for numeri
cal integration—might give you an incorrect answer. The possibility of
this occurring is extremely remote. The algorithm has been designed to
give accurate results with almost any smooth function. Only for func
tions that exhibit extremely erratic behavior is there any substantial
risk of obtaining an inaccurate answer. Such functions rarely occur in
problems related to actual physical situations; when they do, they
usually can be recognized and dealt with in a straightforward manner.
Unfortunately, since all that the algorithm knows about f(x) are its
values at the sample points, it cannot distinguish between f(x) and any
other function that agrees with f(x) at all the sample points. This situa
tion is depicted below, showing (over a portion of the interval of
integration) three functions whose graphs include the many sample
points in common.
274 D: More About Integration