Appendix E: A Detailed Look at f 241
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 algorithm terminates, placing 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.
†
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.
Accuracy, Uncertainty, and Calculation Time
The accuracy of an f approximation does not always change when you
increase by just one the number of digits specified in the display format,
though the uncertainty will decrease. Similarly, the time required to
calculate an integral sometimes changes when you change the display
format, but sometimes does not.
Example: The Bessel function of the first kind, of order four, can be
expressed as
J
4
(x) =
1
0
cos (4θ − x sin θ) dθ.
*
The relationship between the display format, the uncertainly in the function, and the
uncertainty in the approximation to its integral are discussed later in this appendix.
†
Provided that f(x) does not vary rapidly, a consideration that will be discussed in more
detail later in this appendix.
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