Manual HP 32sII (page 332 of 376) (English)

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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,

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HP 32sII Manual

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