Appendix D: A Detailed Look at _ 223
If a calculation has a result whose magnitude is smaller than
1.000000000×10
-99
, the result is set equal to zero. This effect is referred to
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
x
4
= 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
l / x
2
= 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
-
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
| x
2
5 | = 0
has a root at x =
, the
result of using _ is Error 8 (for any initial estimates) because the
function never equals zero nor changes sign. On the other hand, the
equation
[(|x| + 1) + 10
15
]
2
= 10
30
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
f(x) = [(|x| + 1) + 10
15
]
2
- 10
30
,
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