246 Appendix E: A Detailed Look at f
If f (x) relates to a physical situation, then the function you would like to
integrate is not f(x) but rather
F (x) = f (x) ± δ
2
(x),
where δ
2
(x) is the uncertainty associated with f (x) that is caused by the
approximation to the actual physical situation.
Since f (x) = (x)
±
δ
1
(x), the function you want to integrate is
F (x) = (x) ± δ
1
(x) ± δ
2
(x)
or F (x) = (x) ± δ(x),
where δ(x) is the net uncertainty associated with (x).
Therefore, the integral you want is
b
a
F (x) dx =
b
a
[(x)
± δ(x)] dx
=
b
a
(x) dx
±
b
a
δ(x) dx
= I ± Δ
where I is the approximation to
b
a
F (x) dx and ∆ is the uncertainty
associated with the approximation. The f algorithm places the number
I in the X-register and the number ∆ in the Y-register.
The uncertainty δ(x) of (x), the function calculated by your subroutine,
is determined as follows. Suppose you consider three significant digits of
the function’s values to be accurate, so you set the display format to i
2. The display would then show only the accurate digits in the mantissa of
a function’s values: for example,
1.23 -04.
Since the display format rounds the number in the X-register to the
number displayed, this implies that the uncertainty in the function’s
values is
± 0.005 × 10
−4
= ± 0.5 × 10
−2
× 10
−4
= ± 0.5 × 10
−6
.
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