HP 15c - Manual (Page 244)

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.
Keystrokes Display
´ 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.