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-{"documenter":{"julia_version":"1.10.3","generation_timestamp":"2024-05-29T04:49:02","documenter_version":"1.4.1"}}
+{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-07-06T11:07:24","documenter_version":"1.5.0"}}
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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>How ForwardDiff Works · ForwardDiff</title><meta name="title" content="How ForwardDiff Works · ForwardDiff"/><meta property="og:title" content="How ForwardDiff Works · ForwardDiff"/><meta property="twitter:title" content="How ForwardDiff Works · ForwardDiff"/><meta name="description" content="Documentation for ForwardDiff."/><meta property="og:description" content="Documentation for ForwardDiff."/><meta property="twitter:description" content="Documentation for ForwardDiff."/><script data-outdated-warner src="../../assets/warner.js"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="../../">ForwardDiff</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="../../">Introduction</a></li><li><span class="tocitem">User Documentation</span><ul><li><a class="tocitem" href="../../user/limitations/">Limitations of ForwardDiff</a></li><li><a class="tocitem" href="../../user/api/">Differentiation API</a></li><li><a class="tocitem" href="../../user/advanced/">Advanced Usage Guide</a></li><li><a class="tocitem" href="../../user/upgrade/">Upgrading from Older Versions</a></li></ul></li><li><span class="tocitem">Developer Documentation</span><ul><li class="is-active"><a class="tocitem" href>How ForwardDiff Works</a><ul class="internal"><li><a class="tocitem" href="#Dual-Number-Implementation"><span>Dual Number Implementation</span></a></li><li><a class="tocitem" href="#ForwardDiff&#39;s-API"><span>ForwardDiff&#39;s API</span></a></li></ul></li><li><a class="tocitem" href="../contributing/">How to Contribute</a></li></ul></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><a class="is-disabled">Developer Documentation</a></li><li class="is-active"><a href>How ForwardDiff Works</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>How ForwardDiff Works</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/JuliaDiff/ForwardDiff.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/JuliaDiff/ForwardDiff.jl/blob/master/docs/src/dev/how_it_works.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="How-ForwardDiff-Works"><a class="docs-heading-anchor" href="#How-ForwardDiff-Works">How ForwardDiff Works</a><a id="How-ForwardDiff-Works-1"></a><a class="docs-heading-anchor-permalink" href="#How-ForwardDiff-Works" title="Permalink"></a></h1><p>ForwardDiff is an implementation of <a href="https://en.wikipedia.org/wiki/Automatic_differentiation">forward mode automatic differentiation</a> (AD) in Julia. There are two key components of this implementation: the <code>Dual</code> type, and the API.</p><h2 id="Dual-Number-Implementation"><a class="docs-heading-anchor" href="#Dual-Number-Implementation">Dual Number Implementation</a><a id="Dual-Number-Implementation-1"></a><a class="docs-heading-anchor-permalink" href="#Dual-Number-Implementation" title="Permalink"></a></h2><p>Partial derivatives are stored in the <code>Partials</code> type:</p><pre><code class="language-julia hljs">struct Partials{N,V} &lt;: AbstractVector{V}
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>How ForwardDiff Works · ForwardDiff</title><meta name="title" content="How ForwardDiff Works · ForwardDiff"/><meta property="og:title" content="How ForwardDiff Works · ForwardDiff"/><meta property="twitter:title" content="How ForwardDiff Works · ForwardDiff"/><meta name="description" content="Documentation for ForwardDiff."/><meta property="og:description" content="Documentation for ForwardDiff."/><meta property="twitter:description" content="Documentation for ForwardDiff."/><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/catppuccin-mocha.css" data-theme-name="catppuccin-mocha"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/catppuccin-macchiato.css" data-theme-name="catppuccin-macchiato"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><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="../../">ForwardDiff</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="../../">Introduction</a></li><li><span class="tocitem">User Documentation</span><ul><li><a class="tocitem" href="../../user/limitations/">Limitations of ForwardDiff</a></li><li><a class="tocitem" href="../../user/api/">Differentiation API</a></li><li><a class="tocitem" href="../../user/advanced/">Advanced Usage Guide</a></li><li><a class="tocitem" href="../../user/upgrade/">Upgrading from Older Versions</a></li></ul></li><li><span class="tocitem">Developer Documentation</span><ul><li class="is-active"><a class="tocitem" href>How ForwardDiff Works</a><ul class="internal"><li><a class="tocitem" href="#Dual-Number-Implementation"><span>Dual Number Implementation</span></a></li><li><a class="tocitem" href="#ForwardDiff&#39;s-API"><span>ForwardDiff&#39;s API</span></a></li></ul></li><li><a class="tocitem" href="../contributing/">How to Contribute</a></li></ul></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><a class="is-disabled">Developer Documentation</a></li><li class="is-active"><a href>How ForwardDiff Works</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>How ForwardDiff Works</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/JuliaDiff/ForwardDiff.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/JuliaDiff/ForwardDiff.jl/blob/master/docs/src/dev/how_it_works.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="How-ForwardDiff-Works"><a class="docs-heading-anchor" href="#How-ForwardDiff-Works">How ForwardDiff Works</a><a id="How-ForwardDiff-Works-1"></a><a class="docs-heading-anchor-permalink" href="#How-ForwardDiff-Works" title="Permalink"></a></h1><p>ForwardDiff is an implementation of <a href="https://en.wikipedia.org/wiki/Automatic_differentiation">forward mode automatic differentiation</a> (AD) in Julia. There are two key components of this implementation: the <code>Dual</code> type, and the API.</p><h2 id="Dual-Number-Implementation"><a class="docs-heading-anchor" href="#Dual-Number-Implementation">Dual Number Implementation</a><a id="Dual-Number-Implementation-1"></a><a class="docs-heading-anchor-permalink" href="#Dual-Number-Implementation" title="Permalink"></a></h2><p>Partial derivatives are stored in the <code>Partials</code> type:</p><pre><code class="language-julia hljs">struct Partials{N,V} &lt;: AbstractVector{V}
     values::NTuple{N,V}
 end</code></pre><p>The <code>Partials</code> type is used to implement the <code>Dual</code> type:</p><pre><code class="language-julia hljs">struct Dual{T,V&lt;:Real,N} &lt;: Real
     value::V
@@ -43,4 +43,4 @@
                         x_4 + \epsilon_2
                      \end{bmatrix}
 \to
-f(\vec{x}_{\epsilon}) = f(\vec{x}) + \frac{\delta f(\vec{x})}{\delta x_3} \epsilon_1 + \frac{\delta f(\vec{x})}{\delta x_4} \epsilon_2\]</p><p>This seeding process is similar for Jacobians, so we won&#39;t rehash it here.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../user/upgrade/">« Upgrading from Older Versions</a><a class="docs-footer-nextpage" href="../contributing/">How to Contribute »</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="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Wednesday 29 May 2024 04:49">Wednesday 29 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+f(\vec{x}_{\epsilon}) = f(\vec{x}) + \frac{\delta f(\vec{x})}{\delta x_3} \epsilon_1 + \frac{\delta f(\vec{x})}{\delta x_4} \epsilon_2\]</p><p>This seeding process is similar for Jacobians, so we won&#39;t rehash it here.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../user/upgrade/">« Upgrading from Older Versions</a><a class="docs-footer-nextpage" href="../contributing/">How to Contribute »</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="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Saturday 6 July 2024 11:07">Saturday 6 July 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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		{"documenter":{"julia_version":"1.10.3","generation_timestamp":"2024-05-29T04:49:02","documenter_version":"1.4.1"}}
		{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-07-06T11:07:24","documenter_version":"1.5.0"}}