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 <html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Convergence · RationalFunctionApproximation.jl</title><meta name="title" content="Convergence · RationalFunctionApproximation.jl"/><meta property="og:title" content="Convergence · RationalFunctionApproximation.jl"/><meta property="twitter:title" content="Convergence · 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/convergence/"/><meta property="twitter:url" content="https://complexvariables.github.io/RationalFunctionApproximation.jl/convergence/"/><link rel="canonical" 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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><a class="tocitem" href="../">Walkthrough</a></li><li class="is-active"><a class="tocitem" href>Convergence</a></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>Convergence</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Convergence</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/convergence.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="Convergence-of-AAA-rational-approximations"><a class="docs-heading-anchor" href="#Convergence-of-AAA-rational-approximations">Convergence of AAA rational approximations</a><a id="Convergence-of-AAA-rational-approximations-1"></a><a class="docs-heading-anchor-permalink" href="#Convergence-of-AAA-rational-approximations" title="Permalink"></a></h1><pre><code class="language-julia hljs">using RationalFunctionApproximation, CairoMakie</code></pre><p>For a function that is analytic on its domain, the AAA algorithm typically converges at least root-exponentially. In order to observe the convergence, we can construct an approximation that preserves the history of its construction. For example, we approximate an oscillatory function over <span>$[-1,1]$</span> via:</p><pre><code class="language-julia hljs">f = x -&gt; cos(11x)
 r = approximate(f, unit_interval, stats=true)
-convergenceplot(r)</code></pre><img src="042bef55.png" alt="Example block output"/><p>In the plot above, the markers show the estimated max-norm error of the <span>$n$</span>-point AAA rational interpolant over the domain as a function of the numerator/denominator degree <span>$n-1$</span>. The red circles indicate that a pole of the rational interpolant lies on the interval itself. We can verify this by using <code>rewind</code> to recover the degree-7 approximation that was found along the way to <code>r</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; r7 = rewind(r, 7)</code><code class="nohighlight hljs ansi" style="display:block;">Barycentric rational function of type (7,7) on the domain: Path with 1 curve</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; poles(r7)</code><code class="nohighlight hljs ansi" style="display:block;">7-element Vector{ComplexF64}:
- -0.7865385850382729 + 0.26769746910591075im
- -0.7865385850382728 - 0.26769746910591075im
- -0.3174146424804872 + 0.288943809847393im
- -0.3174146424804871 - 0.28894380984739293im
- 0.16129605360549762 - 0.24967968490231618im
- 0.16129605360549765 + 0.2496796849023162im
-  0.7365067113018722 + 0.0im</code></pre><p>When the AAA iteration encounters an approximation with such undesired poles, or having less accuracy than a predecessor, the AAA iteration simply disregards that approximation and continues–-unless there have been more than a designated number of consecutive failures, at which the best interpolant ever encountered is returned. That interpolant is indicated by the gold halo in the convergence plot above.</p><p>When a singularity is very close to the approximation domain, it can cause stagnation and a large number of bad-pole failures:</p><pre><code class="language-julia hljs">f = x -&gt; tanh(3000*(x - 1/5))
+convergenceplot(r)</code></pre><img src="57016ed7.png" alt="Example block output"/><p>In the plot above, the markers show the estimated max-norm error of the <span>$n$</span>-point AAA rational interpolant over the domain as a function of the numerator/denominator degree <span>$n-1$</span>. The red circles indicate that a pole of the rational interpolant lies on the interval itself. We can verify this by using <code>rewind</code> to recover the degree-7 approximation that was found along the way to <code>r</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; r7 = rewind(r, 7)</code><code class="nohighlight hljs ansi" style="display:block;">Barycentric rational function of type (7,7) on the domain: Path with 1 curve</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; poles(r7)</code><code class="nohighlight hljs ansi" style="display:block;">7-element Vector{ComplexF64}:
+ -0.7865385850382719 + 0.2676974691059106im
+ -0.7865385850382718 - 0.2676974691059106im
+  -0.317414642480487 - 0.28894380984739215im
+  -0.317414642480487 + 0.2889438098473921im
+  0.1612960536054982 - 0.2496796849023163im
+ 0.16129605360549823 + 0.24967968490231632im
+  0.7365067113018724 + 0.0im</code></pre><p>When the AAA iteration encounters an approximation with such undesired poles, or having less accuracy than a predecessor, the AAA iteration simply disregards that approximation and continues–-unless there have been more than a designated number of consecutive failures, at which the best interpolant ever encountered is returned. That interpolant is indicated by the gold halo in the convergence plot above.</p><p>When a singularity is very close to the approximation domain, it can cause stagnation and a large number of bad-pole failures:</p><pre><code class="language-julia hljs">f = x -&gt; tanh(3000*(x - 1/5))
 r = approximate(f, unit_interval, stats=true)
-convergenceplot(r)</code></pre><img src="c7709f12.png" alt="Example block output"/><p>This effect is thought to be mainly due to roundoff and conditioning of the problem. If we use more accurate floating-point arithmetic, we can see that the AAA convergence continues steadily  past the previous plateau:</p><pre><code class="language-julia hljs">using DoubleFloats
+convergenceplot(r)</code></pre><img src="2c8a255b.png" alt="Example block output"/><p>This effect is thought to be mainly due to roundoff and conditioning of the problem. If we use more accurate floating-point arithmetic, we can see that the AAA convergence continues steadily  past the previous plateau:</p><pre><code class="language-julia hljs">using DoubleFloats
 r = approximate(f, unit_interval, float_type=Double64, stats=true)
 convergenceplot(r)</code></pre><img src="aa70c007.png" alt="Example block output"/><p>In the extreme case of a function with a singularity on the domain, the convergence can be substantially affected:</p><pre><code class="language-julia hljs">f = x -&gt; abs(x - 1/8)
 r = approximate(f, unit_interval, stats=true)
-convergenceplot(r)</code></pre><img src="68d6eb68.png" alt="Example block output"/><p>In such a case, we might get improvement by increasing the number of allowed consecutive failures via the <code>lookahead</code> keyword argument:</p><pre><code class="language-julia hljs">r = approximate(f, unit_interval, stats=true, lookahead=20)
-convergenceplot(r)</code></pre><img src="6e37860c.png" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Walkthrough</a><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 1.1.0 on <span class="colophon-date" title="Thursday 5 October 2023 18:18">Thursday 5 October 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+convergenceplot(r)</code></pre><img src="f036fc9f.png" alt="Example block output"/><p>In such a case, we might get improvement by increasing the number of allowed consecutive failures via the <code>lookahead</code> keyword argument:</p><pre><code class="language-julia hljs">r = approximate(f, unit_interval, stats=true, lookahead=20)
+convergenceplot(r)</code></pre><img src="10298161.png" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Walkthrough</a><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 1.1.0 on <span class="colophon-date" title="Thursday 5 October 2023 18:36">Thursday 5 October 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>