The ``Big-Oh'' notation is convenient for describing errors.
If and are functions defined for small positive values of ,
we write to mean that there are positive constants and
such that
Since we don't care what the constant is, we can write ``equations'' such as . This is shorthand for the following: We have one function , i.e. when is small, and another function , i.e. when is small. Then , since when is small.
Typically our will represent an error, which of course we want to be as small as possible. The way we could make it small is to use a small . Generally, we might expect a higher order approximation to be better than a lower order one. This is certainly true ``eventually'', i.e. for sufficiently small values of . However, it's not necessarily true for any particular value of . For example, if and , then we might expect when is small. But we might have, say, and . It's true that when is ``small'', but in this case ``small'' means .
Consider the approximation of by
.
This has a Taylor series about
(where we define
).
The easy way to obtain the first few terms of the series is by writing
:
the Taylor series for about starts
, so
The error in the approximation is
Here is a graph showing the error as a function of in the case , using some 34 different values of from to :
<img src="bigoh.gif">