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Pade

Calculates Padé approximant coefficients given sufficient Taylor series coefficients.

https://github.com/aadler/pade

Science Score: 44.0%

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Last synced: 7 months ago · JSON representation ·

Repository

Calculates Padé approximant coefficients given sufficient Taylor series coefficients.

Basic Info
  • Host: GitHub
  • Owner: aadler
  • License: other
  • Language: R
  • Default Branch: master
  • Homepage:
  • Size: 93.8 KB
Statistics
  • Stars: 2
  • Watchers: 0
  • Forks: 1
  • Open Issues: 0
  • Releases: 6
Created over 6 years ago · Last pushed 9 months ago
Metadata Files
Readme Contributing License Code of conduct Citation Security

README.md

title: Package Padé

CRAN Latest Version & Release R-CMD-check Codecov test coverage OpenSSF Best Practices <!-- badges: end -->

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, (PL(x)) of order (L) and (QM(x)) of order (M), such that (\frac{PL(x)}{QM(x)}), 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 aj x^j \end{equation}

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

\begin{equation} A(x) - \frac{PL(x)}{QM(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} &a0 &= p0\ &a1 + a0q1 &= p1\ &a2 + a1q1 + a0q2 &= p2\ &a3 + a2q1 + a1q2 + a0q3 &= p3\ &a4 + a3q1 + a2q2 + a1q3 + a0q4 &= p4\ &\vdots&\vdots\ &a{L+M} + a{L+M-1}q1 + \ldots + a0q{L+M} &= p{L+M} \end{align} ]

remembering that all (pk, k > L) and (qk, 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 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.

Owner

  • Name: Avraham Adler
  • Login: aadler
  • Kind: user
  • Location: New York Metropolitan Area
  • Company: McGill and Partners

Citation (CITATION.cff) 

# --------------------------------------------
# CITATION file created with {cffr} R package
# See also: https://docs.ropensci.org/cffr/
# --------------------------------------------
 
cff-version: 1.2.0
message: 'To cite package "Pade" in publications use:'
type: software
license:
- BSD-2-Clause
- GPL-2.0-or-later
title: 'Pade: Padé Approximant Coefficients'
version: 1.0.8
doi: 10.32614/CRAN.package.Pade
identifiers:
- type: doi
  value: 10.32614/CRAN.package.Pade
abstract: Given a vector of Taylor series coefficients of sufficient length as input,
  the function returns the numerator and denominator coefficients for the Padé approximant
  of appropriate order (Baker, 1975) <ISBN:9780120748556>.
authors:
- family-names: Adler
  given-names: Avraham
  email: Avraham.Adler@gmail.com
  orcid: https://orcid.org/0000-0002-3039-0703
preferred-citation:
  type: manual
  title: 'Pade: Padé Approximant Coefficients'
  authors:
  - family-names: Adler
    given-names: Avraham
    email: Avraham.Adler@gmail.com
    orcid: https://orcid.org/0000-0002-3039-0703
  year: '2015'
  url: https://CRAN.R-project.org/package=Pade
  doi: 10.32614/CRAN.package.Pade
  notes: R package version 1.0.8
repository: https://CRAN.R-project.org/package=Pade
repository-code: https://github.com/aadler/Pade
url: https://github.com/aadler/Pade
date-released: '2025-07-10'
contact:
- family-names: Adler
  given-names: Avraham
  email: Avraham.Adler@gmail.com
  orcid: https://orcid.org/0000-0002-3039-0703

GitHub Events

Total
  • Release event: 1
  • Watch event: 1
  • Push event: 1
Last Year
  • Release event: 1
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Committers

Last synced: over 2 years ago

All Time
  • Total Commits: 90
  • Total Committers: 2
  • Avg Commits per committer: 45.0
  • Development Distribution Score (DDS): 0.322
Past Year
  • Commits: 18
  • Committers: 1
  • Avg Commits per committer: 18.0
  • Development Distribution Score (DDS): 0.0
Top Committers
Name Email Commits
Avraham Adler a****r@g****m 61
Avraham Adler A****r@g****m 29

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Last synced: 8 months ago

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Packages

  • Total packages: 1
  • Total downloads:
    • cran 375 last-month
  • Total docker downloads: 43,390
  • Total dependent packages: 0
  • Total dependent repositories: 0
  • Total versions: 13
  • Total maintainers: 1
cran.r-project.org: Pade

Padé Approximant Coefficients

  • Versions: 13
  • Dependent Packages: 0
  • Dependent Repositories: 0
  • Downloads: 375 Last month
  • Docker Downloads: 43,390
Rankings
Forks count: 28.8%
Dependent packages count: 29.8%
Average: 32.5%
Downloads: 33.3%
Stargazers count: 35.2%
Dependent repos count: 35.5%
Maintainers (1)
Avraham.Adler@gmail.com
Last synced: 8 months ago

Dependencies

DESCRIPTION cran
  • utils * imports
  • covr * suggests
  • tinytest * suggests
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.github/workflows/test-coverage.yaml actions
  • actions/checkout v3 composite
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  • r-lib/actions/setup-r v2 composite
  • r-lib/actions/setup-r-dependencies v2 composite
.github/workflows/rhub.yaml actions
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