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D–2 More about Integration
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24 Size : 17.7 x 25.2 cm
As explained in chapter 8, the uncertainty of the final approximation is a
number derived from the display format, which specifies the uncertainty for
the function. At the end of each iteration, the algorithm compares the
approximation calculated during that iteration with the approximations
calculated during two previous iterations. If the difference between any of
these three approximations and the other two 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.
It is extremely unlikely that the errors in each of three successive
approximations — 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
approximation will be less than its uncertainty (provided that f(x) does not
vary rapidly). Although we can't know the error in the final approximation,
the error is extremely unlikely to exceed the displayed uncertainty of the
approximation. In other words, the uncertainty estimate 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 32SII is one of the best available,
in certain situations it — like all other algorithms for numerical
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 functions 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 ire 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 situation is depicted below,