Table of Contents
Advertisement
A More Detailed Look at (73]
237
previous iterations. If the difference between any of these three approxi-
mations and the other two is less than the uncertainty of the final approx-
imation, the algorithm terminates, placing the current approximation in
the X-register and its uncertainty in the Y-register.
The
algorithm is designed so that it is extremely unlikely that the
error in each of three successive approximations—thatis, the differences
between the actual integral and the approximations—would all be less
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, we can
be very confident that the error is less than the displayed uncertainty of
the approximation. Thus, the uncertainty of the approximation is an
"'upper bound'' on the difference between the approximation and the
actual integral.
Accuracy, Uncertainty, and Calculation Time
The accuracy of an
approximation does not always change when you
increase by just one the number of digits specified in the display format.
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
Ji(x) = %J.o cos (46 — xsin6) df.
Calculate the integral in the expression for J(1),
1
m
Hf
cos (46 — sin6) do.
T
0
* Provided thatf(x) is sufficiently smooth, a consideration we will discuss in more detail later
in this appendix.
Advertisement
Table of Contents
