## Mathematics

### Introduction

For a given function, its 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 (), called the Padé approximant
of order ([L/M]), “agrees” with the original function in order (L + M).
More precisely, given

\[\begin{equation}
A(x) = \sum_{j=0}^\infty a_j x^j
\end{equation}\]

the Padé approximant of order ([L/M]) to (A(x)) has the property
that

\[\begin{equation}
A(x) - \frac{P_L(x)}{Q_M(x)} = \mathcal{O}\left(x^{L + M + 1}\right)
\end{equation}\]

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:

[ \[\begin{align}
&a_0 &= p_0\\
&a_1 + a_0q_1 &= p_1\\
&a_2 + a_1q_1 + a_0q_2 &= p_2\\
&a_3 + a_2q_1 + a_1q_2 + a_0q_3 &= p_3\\
&a_4 + a_3q_1 + a_2q_2 + a_1q_3 + a_0q_4 &= p_4\\
&\vdots&\vdots\\
&a_{L+M} + a_{L+M-1}q_1 + \ldots + a_0q_{L+M} &= p_{L+M}
\end{align}\] ]

remembering that all (p_k, k > L) and (q_k, k > M) are 0.

Given integers `L`

and `M`

, and vector
`A`

, a vector of Taylor series coefficients, in increasing
order and length at least `L + M + 1`

, the `Pade`

function returns a list of two elements, `Px`

and
`Qx`

, which are the coefficients of the Padé approximant
numerator and denominator respectively, in increasing order.

## Citation

If you use the package, please cite it per CITATION.

## Contributions

Please see CONTRIBUTING.md.

## Roadmap

### Major

- There are no plans for major changes at this time

### Minor

- There are no plans for minor changes at this time

## Security

Please see SECURITY.md.