Appendix E: A Detailed Look at f 245
This approximation took about twice as long as the approximation in i
3 or i 2. In this case, the algorithm had to evaluate the function at
about twice as many sample points as before in order to achieve an
approximation of acceptable accuracy. Note, however, that you received
a reward for your patience: the accuracy of this approximation is better,
by almost two digits, than the accuracy of the approximation calculated
using half the number of sample points.
The preceding examples show that repeating the approximation of an
integral in a different display format sometimes will give you a more
accurate answer, but sometimes it will not. Whether or not the accuracy
is changed depends on the particular function, and generally can be
determined only by trying it.
Furthermore, if you do get a more accurate answer, it will come at the cost
of about double the calculation time. This unavoidable trade-off between
accuracy and calculation time is important to keep in mind if you are
considering decreasing the uncertainty in hopes of obtaining a more
accurate answer.
The time required to calculate the integral of a given function depends not
only on the number of digits specified in the display format, but also, to a
certain extent on the limits of integration. When the calculation of an
integral requires an excessive amount of time, the width of the interval of
integration (that is, the difference of the limits) may be too large
compared with certain features of the function being integrated. For most
problems, however, you need not be concerned about the effects of the
limits of integration on the calculation time. These conditions, as well as
techniques for dealing with such situations, will be discussed later in this
appendix.
Uncertainty and the Display Format
Because of round-off error, the subroutine you write for evaluating f(x)
cannot calculate f(x) exactly, but rather calculates
(x) = f (x) ± δ
1
(x),
where δ
1
(x) is the uncertainty of f (x) caused by round-off error.
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