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Padé

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 (\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 ]

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) ]

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.

Function Input and Output

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 as:

Avraham Adler (2015). Pade: Padé Approximant Coefficients. R package version 1.0.1. https://CRAN.R-project.org/package=Pade

A BibTeX entry for LaTeX users is:

  @Manual{,
    title = {Pade: Padé Approximant Coefficients},
    author = {Avraham Adler},
    year = {2015},
    note = {R package version 1.0.1},
    url = {https://CRAN.R-project.org/package=Pade},
  }

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Tests

Please review the current test suite and supply similar testthat-compatible unit tests for all added functionality.

Submission

If you would like to contribute to the project, it may be prudent to first contact the maintainer via email. A request or suggestion may be raised as an issue as well. To supply a pull request (PR), please:

1. Fork the project and then clone into your own local repository
2. Create a branch in your repository in which you will make your changes
3. Push that branch and then create a pull request

At this point, the PR will be discussed and eventually accepted or rejected.