How f works, Accuracy, uncertainty, and calculation time – HP 15c User Manual
Page 241
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
π
dθ
θ
x
θ
x
J
0
4
sin
4
cos
1
)
(
*
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.