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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>Introduction · ImplicitDifferentiation.jl</title><script data-outdated-warner src="../../assets/warner.js"></script><link rel="canonical" href="https://gdalle.github.io/ImplicitDifferentiation.jl/examples/0_intro/"/><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/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/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="../../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="../../">ImplicitDifferentiation.jl</a></span></div><form class="docs-search" action="../../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../../">Home</a></li><li><a class="tocitem" href="../../faq/">FAQ</a></li><li><span class="tocitem">Examples</span><ul><li class="is-active"><a class="tocitem" href>Introduction</a><ul class="internal"><li><a class="tocitem" href="#Why-do-we-bother?"><span>Why do we bother?</span></a></li><li><a class="tocitem" href="#Implicit-function"><span>Implicit function</span></a></li><li><a class="tocitem" href="#Forward-and-reverse-mode-autodiff"><span>Forward and reverse mode autodiff</span></a></li><li><a class="tocitem" href="#Second-derivative"><span>Second derivative</span></a></li></ul></li><li><a class="tocitem" href="../1_basic/">Basic use cases</a></li><li><a class="tocitem" href="../2_advanced/">Advanced use cases</a></li><li><a class="tocitem" href="../3_tricks/">Tricks</a></li></ul></li><li><a class="tocitem" href="../../api/">API reference</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"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>Introduction</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Introduction</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/gdalle/ImplicitDifferentiation.jl/blob/c5847dc25f8f51d42305c43c91892f52d2e2afa1/examples/0_intro.jl#" title="View on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">View on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Introduction"><a class="docs-heading-anchor" href="#Introduction">Introduction</a><a id="Introduction-1"></a><a class="docs-heading-anchor-permalink" href="#Introduction" title="Permalink"></a></h1><p>We explain the basics of our package on a simple function that is not amenable to naive automatic differentiation.</p><pre><code class="language-julia hljs">using ForwardDiff
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Introduction · ImplicitDifferentiation.jl</title><script data-outdated-warner src="../../assets/warner.js"></script><link rel="canonical" href="https://gdalle.github.io/ImplicitDifferentiation.jl/examples/0_intro/"/><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/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/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="../../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="../../">ImplicitDifferentiation.jl</a></span></div><form class="docs-search" action="../../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../../">Home</a></li><li><a class="tocitem" href="../../faq/">FAQ</a></li><li><span class="tocitem">Examples</span><ul><li class="is-active"><a class="tocitem" href>Introduction</a><ul class="internal"><li><a class="tocitem" href="#Why-do-we-bother?"><span>Why do we bother?</span></a></li><li><a class="tocitem" href="#Implicit-function"><span>Implicit function</span></a></li><li><a class="tocitem" href="#Forward-and-reverse-mode-autodiff"><span>Forward and reverse mode autodiff</span></a></li><li><a class="tocitem" href="#Second-derivative"><span>Second derivative</span></a></li></ul></li><li><a class="tocitem" href="../1_basic/">Basic use cases</a></li><li><a class="tocitem" href="../2_advanced/">Advanced use cases</a></li><li><a class="tocitem" href="../3_tricks/">Tricks</a></li></ul></li><li><a class="tocitem" href="../../api/">API reference</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"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>Introduction</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Introduction</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/gdalle/ImplicitDifferentiation.jl/blob/50deafd521e08164a1e3f5788316d0ca8dc3444b/examples/0_intro.jl#" title="View on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">View on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Introduction"><a class="docs-heading-anchor" href="#Introduction">Introduction</a><a id="Introduction-1"></a><a class="docs-heading-anchor-permalink" href="#Introduction" title="Permalink"></a></h1><p>We explain the basics of our package on a simple function that is not amenable to naive automatic differentiation.</p><pre><code class="language-julia hljs">using ForwardDiff
using ImplicitDifferentiation
using LinearAlgebra
using Random
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forward, conditions; linear_solver=DirectLinearSolver()
)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">ImplicitFunction(forward, conditions, DirectLinearSolver(true), nothing)</code></pre><p>Then the Jacobian itself is differentiable.</p><pre><code class="language-julia hljs">h = rand(2)
J_Z(t) = Zygote.jacobian(implicit_higher_order, x .+ t .* h)[1]
ForwardDiff.derivative(J_Z, 0) ≈ Diagonal((-0.25 .* h) ./ (x .^ 1.5))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">true</code></pre><hr/><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../faq/">« FAQ</a><a class="docs-footer-nextpage" href="../1_basic/">Basic use cases »</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></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Thursday 10 August 2023 08:56">Thursday 10 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
ForwardDiff.derivative(J_Z, 0) ≈ Diagonal((-0.25 .* h) ./ (x .^ 1.5))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">true</code></pre><hr/><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../faq/">« FAQ</a><a class="docs-footer-nextpage" href="../1_basic/">Basic use cases »</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></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Thursday 10 August 2023 09:07">Thursday 10 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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