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

J4 (x) = π1 ∫0π cos(4θ − xsinθ)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.