244 Appendix E: A Detailed Look at f
All 10 digits of the approximations in i 2 and i 3 are identical: the
accuracy of the approximation in i 3 is no better than the accuracy in
i 2 despite the fact that the uncertainty in i 3 is less than the
uncertainty in i 2. Why is this? Remember that the accuracy of any
approximation depends primarily on the number of sample points at
which the function f(x) has been evaluated. The f algorithm is iterated
with increasing numbers of sample points until the disparity among three
successive approximations is less than the uncertainty derived from the
display format. After a particular iteration, the disparity among the
approximations may already be so much less than the uncertainty that it
would still be less if the uncertainty were decreased by a factor of 10. In
such cases, if you decreased the uncertainty by specifying one more digit
in the display format, the algorithm would not have to consider additional
sample points, and the resulting approximation would be identical to the
approximation calculated with the larger uncertainty.
If you calculated the two preceding approximations on your calculator,
you may have noticed that it did not take any longer to calculate the
integral in i 3 than in i 2. This is because the time to calculate the
integral of a given function depends on the number of sample points at
which the function must be evaluated to achieve an approximation of
acceptable accuracy. For the i 3 approximation, the algorithm did not
have to consider more sample points than it did in i 2, so it did not
take any longer to calculate the integral.
Often, however, increasing the number of digits in the display format will
require evaluating the function at additional sample points, so that
calculating the integral will take more time. Now calculate the same
integral in i 4.
´ i 4
7.7858 -03
i 4 display.
) )
3.1416 00
Rolls down stack until upper
limit appears in X-register.
´ f 0
7.7807 -03
Integral approximated in
i 4.
To Next Page
Loading...
Need help?
General
