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,

D–2

More about Integration

File name 32sii-Manual-E-0424

 

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm