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{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-09-15T18:58:19","documenter_version":"1.0.0"}}
{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-09-21T18:28:50","documenter_version":"1.0.1"}}
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<!DOCTYPE html>
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Home · RationalFunctionApproximation.jl</title><meta name="title" content="Home · RationalFunctionApproximation.jl"/><meta property="og:title" content="Home · RationalFunctionApproximation.jl"/><meta property="twitter:title" content="Home · RationalFunctionApproximation.jl"/><meta name="description" content="Documentation for RationalFunctionApproximation.jl."/><meta property="og:description" content="Documentation for RationalFunctionApproximation.jl."/><meta property="twitter:description" content="Documentation for RationalFunctionApproximation.jl."/><meta property="og:url" content="https://complexvariables.github.io/RationalFunctionApproximation.jl/"/><meta property="twitter:url" content="https://complexvariables.github.io/RationalFunctionApproximation.jl/"/><link rel="canonical" href="https://complexvariables.github.io/RationalFunctionApproximation.jl/"/><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="assets/documenter.js"></script><script src="search_index.js"></script><script src="siteinfo.js"></script><script src="../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href>RationalFunctionApproximation.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>Home</a><ul class="internal"><li><a class="tocitem" href="#Approximation-on-[-1,-1]"><span>Approximation on [-1, 1]</span></a></li><li><a class="tocitem" href="#Approximation-on-other-domains"><span>Approximation on other domains</span></a></li><li><a class="tocitem" href="#Unbounded-domains"><span>Unbounded domains</span></a></li></ul></li><li><a class="tocitem" href="functions/">Functions</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Home</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Home</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/complexvariables/RationalFunctionApproximation.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/complexvariables/RationalFunctionApproximation.jl/blob/main/docs/src/index.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Rational-function-approximation-in-Julia"><a class="docs-heading-anchor" href="#Rational-function-approximation-in-Julia">Rational function approximation in Julia</a><a id="Rational-function-approximation-in-Julia-1"></a><a class="docs-heading-anchor-permalink" href="#Rational-function-approximation-in-Julia" title="Permalink"></a></h1><p><em>Documentation for <a href="https://github.com/complexvariables/RationalFunctionApproximation.jl">RationalFunctionApproximation.jl</a>.</em></p><p>This package uses the continuous form of the AAA algorithm to adaptively compute rational approximations of functions on intervals and other domains in the complex plane. See <a href="https://arxiv.org/abs/2305.03677">AAA rational approximation on a continuum</a>, which is to appear in SISC.</p><h2 id="Approximation-on-[-1,-1]"><a class="docs-heading-anchor" href="#Approximation-on-[-1,-1]">Approximation on [-1, 1]</a><a id="Approximation-on-[-1,-1]-1"></a><a class="docs-heading-anchor-permalink" href="#Approximation-on-[-1,-1]" title="Permalink"></a></h2><p>Here&#39;s a smooth, gentle function on the interval <span>$[-1, 1]$</span>:</p><pre><code class="language-julia hljs">using RationalFunctionApproximation, CairoMakie
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Walkthrough · RationalFunctionApproximation.jl</title><meta name="title" content="Walkthrough · RationalFunctionApproximation.jl"/><meta property="og:title" content="Walkthrough · RationalFunctionApproximation.jl"/><meta property="twitter:title" content="Walkthrough · RationalFunctionApproximation.jl"/><meta name="description" content="Documentation for RationalFunctionApproximation.jl."/><meta property="og:description" content="Documentation for RationalFunctionApproximation.jl."/><meta property="twitter:description" content="Documentation for RationalFunctionApproximation.jl."/><meta property="og:url" content="https://complexvariables.github.io/RationalFunctionApproximation.jl/"/><meta property="twitter:url" content="https://complexvariables.github.io/RationalFunctionApproximation.jl/"/><link rel="canonical" href="https://complexvariables.github.io/RationalFunctionApproximation.jl/"/><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="assets/documenter.js"></script><script src="search_index.js"></script><script src="siteinfo.js"></script><script src="../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href>RationalFunctionApproximation.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>Walkthrough</a><ul class="internal"><li><a class="tocitem" href="#Approximation-on-[-1,-1]"><span>Approximation on [-1, 1]</span></a></li><li><a class="tocitem" href="#Approximation-on-other-domains"><span>Approximation on other domains</span></a></li><li><a class="tocitem" href="#Unbounded-domains"><span>Unbounded domains</span></a></li></ul></li><li><a class="tocitem" href="minimax/">Minimax</a></li><li><a class="tocitem" href="mode/">Discrete vs. continuous</a></li><li><a class="tocitem" href="functions/">Functions</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Walkthrough</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Walkthrough</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/complexvariables/RationalFunctionApproximation.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/complexvariables/RationalFunctionApproximation.jl/blob/main/docs/src/index.md#" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Rational-function-approximation-in-Julia"><a class="docs-heading-anchor" href="#Rational-function-approximation-in-Julia">Rational function approximation in Julia</a><a id="Rational-function-approximation-in-Julia-1"></a><a class="docs-heading-anchor-permalink" href="#Rational-function-approximation-in-Julia" title="Permalink"></a></h1><p><em>Documentation for <a href="https://github.com/complexvariables/RationalFunctionApproximation.jl">RationalFunctionApproximation.jl</a>.</em></p><p>This package uses the continuous form of the AAA algorithm to adaptively compute rational approximations of functions on intervals and other domains in the complex plane. See <a href="https://arxiv.org/abs/2305.03677">AAA rational approximation on a continuum</a>, which is to appear in SISC.</p><h2 id="Approximation-on-[-1,-1]"><a class="docs-heading-anchor" href="#Approximation-on-[-1,-1]">Approximation on [-1, 1]</a><a id="Approximation-on-[-1,-1]-1"></a><a class="docs-heading-anchor-permalink" href="#Approximation-on-[-1,-1]" title="Permalink"></a></h2><p>Here&#39;s a smooth, gentle function on the interval <span>$[-1, 1]$</span>:</p><pre><code class="language-julia hljs">using RationalFunctionApproximation, CairoMakie
const shg = current_figure
f = x -&gt; exp(cos(4x) - sin(3x))
lines(-1..1, f)</code></pre><img src="index-8263853f.png" alt="Example block output"/><p>To create a rational function that approximates <span>$f$</span> well on this domain, we use the continuous form of the AAA algorithm:</p><pre><code class="language-julia hljs">r = aaa(f)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Barycentric function with 20 nodes and values:
Expand Down Expand Up @@ -54,4 +54,4 @@
lines!([(0, 8), (0, -8)], color=:white, linewidth=5)
scatter!(φ.(nodes(r.fun)), color=:black, markersize=10)
limits!(-8, 8, -8, 8)
shg()</code></pre><img src="index-1bb9e9e9.png" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="functions/">Functions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 15 September 2023 18:58">Friday 15 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
shg()</code></pre><img src="index-1bb9e9e9.png" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="minimax/">Minimax »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.1 on <span class="colophon-date" title="Thursday 21 September 2023 18:28">Thursday 21 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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