Chebyshev Series
- For example, we can get the first four terms of a Chebyshev series
by starting with the Maclaurin expansion for
.
- Such a series converges more rapidly than does a Taylor series on
;
- Replacing terms by Eqn. 2, but omitting polynomials beyond
because we want only four terms, we have;
- The number of terms that are employed determines the accuracy of the computed values.
- To compare the Chebyshev expansion with the Maclaurin series, we convert back to powers of
, using Eqn. 1:
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e^x=0.9946+0.9973x+0.5430x^2+0.1772x^3+\ldots
\end{displaymath}" |
(3) |
Table 2:
Comparison of Chebyshev series for
with Maclaurin series.
![\begin{table}\begin{center}
\includegraphics[scale=0.8]{figures/4.3.ps}
\end{center}
\end{table}](img60.png) |
- Table 2 and Figure 2 compare the error of the Chebyshev expansion (
) with the Maclaurin series
.
- Chebyshev expansion, the errors can be considered to be distributed more or less uniformly throughout the interval.
- Maclaurin expansion, which gives very small errors near the origin, allows the error to bunch up at the ends of the interval.
Figure 2:
Comparison of the error of Chebyshev series for
with the error of Maclaurin series.
|
Cem Ozdogan
2010-12-19