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Appendix D: A Detailed Look at _
223
If a calculation has a result whose magnitude is smaller than
-99
1.000000000×10
, the result is set equal to zero. This effect is referred to
as ―underflow.‖ If the subroutine that calculates your function encounters
underflow for a range of x and if this affects the value of the function, then a
root in this range may be expected to have some inaccuracy. For example,
the equation
4
x
= 0
has a root at x = 0. Because of underflow, _ produces a root of
1.5060
-25 (for initial estimates of 1 and 2). As another example,
consider the equation
2
l / x
= 0
whose root is infinite in value. Because of underflow, _ gives a root
of 3.1707
49 (for initial estimates of 10 and 20). In each of these
examples, the algorithm has found a value of x for which the calculated
function value equals zero. By understanding the effect of underflow, you
can readily interpret results such as these.
The accuracy of a computed value sometimes can be adversely affected by
―round-off‖ error, by which an infinitely precise number is rounded to 10
significant digits. If your subroutine requires extra precision to properly
calculate the function for a range of x, the result obtained by _ may
be inaccurate. For example, the equation
2
– 5 | = 0
| x
has a root at x =
. Because no 10-digit number exactly equals
, the
5
5
result of using _ is
(for any initial estimates) because the
Error 8
function never equals zero nor changes sign. On the other hand, the
equation
15
2
30
[(|x| + 1) + 10
]
= 10
has no roots because the left side of the equation is always greater than the
right side. However, because of round-off in the calculation of
15
2
30
f(x) = [(|x| + 1) + 10
]
- 10
,
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