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Closing tag in NEWS, some spacing in README.
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aadler committed Jun 10, 2015
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12 changes: 6 additions & 6 deletions README.md
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For a given function, its The [Taylor series](https://en.wikipedia.org/wiki/Taylor_series) is the "best" polynomial representations of that function. If the function is being evaluated at 0, the Taylor series representation is also called the Maclaurin series. The error is proportional to the first "left-off" term. Also, the series is only a good estimate in a small radius around the point for which it is calculated (e.g. 0 for a Maclaurin series).

Padé approximants estimate functions as the quotient of two polynomials. Specifically, given a Taylor series expansion of a function $T(x)$ of order $L + M$, there are two polynomials, $P_L(x)$ of order $L$ and $Q_M(x)$ of order $M$, such that $\frac{P_L(x)}{Q_M(x)}$, called the Padé approximant of order $[L/M]$, "agrees" with the original function in order $L + M$. More precisely, given
$$
A(x) = \sum_{j=0}^\infty a_j x^j
$$
A(x) = \sum_{j=0}^\infty a_j x^j
$$
the Padé approximant of order $[L/M]$ to $A(x)$ has the property that
$$
A(x) - \frac{P_L(x)}{Q_M(x)} = \mathcal{O}\left(x^{L + M + 1}\right)
$$
A(x) - \frac{P_L(x)}{Q_M(x)} = \mathcal{O}\left(x^{L + M + 1}\right)
$$
The Padé approximant consistently has a wider radius of convergence than its parent Taylor series, often converging where the Taylor series does not. This makes it very suitable for numerical computation.

##Calculation
With the normalization that the first term of $Q(x)$ is always 1, there is a set of linear equations which will generate the unique Padé approximant coefficients. Letting $a_n$ be the coefficients for the Taylor series, one can solve:
$$
$$
\begin{align}
&a_0 &= p_0\\
&a_1 + a_0q_1 &= p_1\\
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&a_4 + a_3q_1 + a_2q_2 + a_1q_3 + a_0q_4 &= p_4\\
&\vdots&\vdots\\
&a_{L+M} + a_{L+M-1}q_1 + \ldots + a_0q_{L+M} &= p_{L+M}
\end{align}
\end{align}
$$
remembering that all $p_k, k > L$ and $q_k, k > M$ are 0.

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9 changes: 5 additions & 4 deletions inst/NEWS.Rd
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\section{Changes in version 0.1-2 (2015-06-10)}{
\subsection{Updates}{
\itemize{
\item Corrected use of grave to proper aigu (Thanks to Dirk Eddelbuettel)
\item Made polynomial order variable names consistent in documentation
\item Minor clarifying prose tweaks to documentation
\itemize{
\item Corrected use of grave to proper aigu (Thanks to Dirk Eddelbuettel)
\item Made polynomial order variable names consistent in documentation
\item Minor clarifying prose tweaks to documentation
}
}
}

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