https://github.com/tpapp/SpectralKit.jl

Building blocks of spectral methods for Julia.
https://github.com/tpapp/SpectralKit.jl

approximation chebyshev julia julia-language package spectral-methods

Last synced: 2 months ago
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Building blocks of spectral methods for Julia.

• Host: GitHub
• URL: https://github.com/tpapp/SpectralKit.jl
• Owner: tpapp
• License: other
• Created: 2019-11-08T10:55:53.000Z (over 5 years ago)
• Default Branch: master
• Last Pushed: 2024-06-18T16:56:23.000Z (about 1 year ago)
• Last Synced: 2024-11-10T00:11:50.215Z (8 months ago)
• Topics: approximation, chebyshev, julia, julia-language, package, spectral-methods
• Language: Julia
• Size: 4.88 MB
• Stars: 24
• Watchers: 3
• Forks: 0
• Open Issues: 4
• Metadata Files:
  • Readme: README.md
  • Changelog: CHANGELOG.md
  • License: LICENSE.md

Awesome Lists containing this project

• awesome-sciml - tpapp/SpectralKit.jl: Building blocks of spectral methods for Julia.

README

        
# SpectralKit.jl

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[![build](https://github.com/tpapp/SpectralKit.jl/workflows/CI/badge.svg)](https://github.com/tpapp/SpectralKit.jl/actions?query=workflow%3ACI)

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[![Documentation](https://img.shields.io/badge/docs-stable-blue.svg)](https://tpapp.github.io/SpectralKit.jl/stable)

[![Documentation](https://img.shields.io/badge/docs-master-blue.svg)](https://tpapp.github.io/SpectralKit.jl/dev)

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[![Aqua QA](https://raw.githubusercontent.com/JuliaTesting/Aqua.jl/master/badge.svg)](https://github.com/JuliaTesting/Aqua.jl)

Building blocks of spectral methods for Julia. Currently includes Chebyshev polynomials on univariate and Smolyak (multivariate) grids, with domain transformations to semi-infinite and infinite domains.

## Introduction

Mostly useful for algorithms along the lines of

> Boyd, John P. *Chebyshev and Fourier spectral methods*. 2001.

The aim is to provide simple, well-tested, robust, and fast *building blocks* for spectral algorithms, which can be easily combined into algorithms.

At the moment, the package API is experimental and subject to change.

## Help

Asking for help in [issues](https://github.com/tpapp/SpectralKit.jl/issues) is fine, you can also ping me as `@Tamas_Papp` on the [Discourse forum](https://discourse.julialang.org/)

## Pretty pictures

Some examples generated this library. **Circles mark values at the limit, shifted horizontally when this is needed to avoid overlap**. Infinite limits shown at finite values, so of course they don't match (this is a visual check of continuity, naturally it is unit tested).

### Chebyshev polynomials and their derivatives





### Chebyshev rational functions on [0,∞)

Up close, you can see the oscillation.



Let's zoom out a bit to see convergence to 0 at ∞.



Derivatives die out faster.



### Chebyshev rational functions on (-∞,∞)

Up close, you can see the oscillation.



Let's zoom out a bit to see convergence at -∞ and ∞.



Derivatives die out slower than for the [0,∞) transformation.



### A Smolyak grid

With `B = 3`.



## Bibliography

- Boyd, J. P. (2001). Chebyshev and fourier spectral methods. Courier Corporation.

- Xu, K. (2016). The chebyshev points of the first kind. Applied Numerical Mathematics, 102, 17–30.