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
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
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
―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
|
x
2
– 5 | = 0
has a root at x =
5
. Because no 10-digit number
exactly
equals
5
, 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
,
Summary of Contents for HP-15C
Page 1: ...HP 15C Owner s Handbook HP Part Number 00015 90001 Edition 2 4 Sep 2011 ...
Page 17: ...Part l HP 15C Fundamentals ...
Page 64: ......
Page 65: ...Part ll HP 15C Programming ...
Page 118: ...118 ...
