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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>LICENSE · InverseFunctions</title><script data-outdated-warner src="../assets/warner.js"></script><link rel="canonical" href="https://JuliaMath.github.io/InverseFunctions.jl/stable/LICENSE/"/><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="../">InverseFunctions</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="../api/">API</a></li><li class="is-active"><a class="tocitem" href>LICENSE</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 class="is-active"><a href>LICENSE</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>LICENSE</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/JuliaMath/InverseFunctions.jl/blob/master/docs/src/LICENSE.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit 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="LICENSE"><a class="docs-heading-anchor" href="#LICENSE">LICENSE</a><a id="LICENSE-1"></a><a class="docs-heading-anchor-permalink" href="#LICENSE" title="Permalink"></a></h1><p>The InverseFunctions.jl package is licensed under the MIT &quot;Expat&quot; License:</p><blockquote><p>Copyright (c) 2021:</p><p>Oliver Schulz &lt;[email protected]&gt; and David Widmann</p><p>Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the &quot;Software&quot;), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:</p><p>The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.</p><p>THE SOFTWARE IS PROVIDED &quot;AS IS&quot;, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.</p></blockquote></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../api/">« API</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="Tuesday 1 August 2023 12:37">Tuesday 1 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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<!DOCTYPE html>
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>API · InverseFunctions</title><script data-outdated-warner src="../assets/warner.js"></script><link rel="canonical" href="https://JuliaMath.github.io/InverseFunctions.jl/stable/api/"/><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="../">InverseFunctions</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 class="is-active"><a class="tocitem" href>API</a><ul class="internal"><li><a class="tocitem" href="#Interface"><span>Interface</span></a></li><li><a class="tocitem" href="#Test-utility"><span>Test utility</span></a></li><li><a class="tocitem" href="#Additional-functionality"><span>Additional functionality</span></a></li></ul></li><li><a class="tocitem" href="../LICENSE/">LICENSE</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 class="is-active"><a href>API</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>API</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/JuliaMath/InverseFunctions.jl/blob/master/docs/src/api.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit 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="API"><a class="docs-heading-anchor" href="#API">API</a><a id="API-1"></a><a class="docs-heading-anchor-permalink" href="#API" title="Permalink"></a></h1><h2 id="Interface"><a class="docs-heading-anchor" href="#Interface">Interface</a><a id="Interface-1"></a><a class="docs-heading-anchor-permalink" href="#Interface" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-binding" id="InverseFunctions.inverse" href="#InverseFunctions.inverse"><code>InverseFunctions.inverse</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">inverse(f)</code></pre><p>Return the inverse of function <code>f</code>.</p><p><code>inverse</code> supports mapped and broadcasted functions (via <code>Base.Broadcast.BroadcastFunction</code> or <code>Base.Fix1</code>) and function composition (requires Julia &gt;= 1.6).</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; foo(x) = inv(exp(-x) + 1);

julia&gt; inv_foo(y) = log(y / (1 - y));

julia&gt; InverseFunctions.inverse(::typeof(foo)) = inv_foo;

julia&gt; InverseFunctions.inverse(::typeof(inv_foo)) = foo;

julia&gt; x = 4.2;

julia&gt; inverse(foo)(foo(x)) ≈ x
true

julia&gt; inverse(inverse(foo)) === foo
true

julia&gt; broadcast_foo = VERSION &gt;= v&quot;1.6&quot; ? Base.Broadcast.BroadcastFunction(foo) : Base.Fix1(broadcast, foo);

julia&gt; X = rand(10);

julia&gt; inverse(broadcast_foo)(broadcast_foo(X)) ≈ X
true

julia&gt; bar = log ∘ foo;

julia&gt; VERSION &lt; v&quot;1.6&quot; || inverse(bar)(bar(x)) ≈ x
true</code></pre><p><strong>Implementation</strong></p><p>Implementations of <code>inverse(::typeof(f))</code> have to satisfy</p><ul><li><code>inverse(f)(f(x)) ≈ x</code> for all <code>x</code> in the domain of <code>f</code>, and</li><li><code>inverse(inverse(f))</code> is defined and <code>inverse(inverse(f))(x) ≈ f(x)</code> for all <code>x</code> in the domain of <code>f</code>.</li></ul><p>You can check your implementation with <a href="#InverseFunctions.test_inverse"><code>InverseFunctions.test_inverse</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/InverseFunctions.jl/blob/d397fef770dd28a78dd0df0d09e94c30b321f4db/src/inverse.jl#L3-L52">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="InverseFunctions.NoInverse" href="#InverseFunctions.NoInverse"><code>InverseFunctions.NoInverse</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct NoInverse{F}</code></pre><p>An instance <code>NoInverse(f)</code> signifies that <code>inverse(f)</code> is not defined.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/InverseFunctions.jl/blob/d397fef770dd28a78dd0df0d09e94c30b321f4db/src/inverse.jl#L58-L62">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="InverseFunctions.setinverse" href="#InverseFunctions.setinverse"><code>InverseFunctions.setinverse</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">setinverse(f, invf)</code></pre><p>Return a function that behaves like <code>f</code> and uses <code>invf</code> as its inverse.</p><p>Useful in cases where no inverse is defined for <code>f</code> or to set an inverse that is only valid within a given context, e.g. only for a limited argument range that is guaranteed by the use case but not in general.</p><p>For example, <code>asin</code> is not a valid inverse of <code>sin</code> for arbitrary arguments of <code>sin</code>, but can be a valid inverse if the use case guarantees that the argument of <code>sin</code> will always be within <code>-π</code> and <code>π</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; foo = setinverse(sin, asin);

julia&gt; x = π/3;

julia&gt; foo(x) == sin(x)
true

julia&gt; inverse(foo)(foo(x)) ≈ x
true

julia&gt; inverse(foo) === setinverse(asin, sin)
true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/InverseFunctions.jl/blob/d397fef770dd28a78dd0df0d09e94c30b321f4db/src/setinverse.jl#L24-L51">source</a></section></article><h2 id="Test-utility"><a class="docs-heading-anchor" href="#Test-utility">Test utility</a><a id="Test-utility-1"></a><a class="docs-heading-anchor-permalink" href="#Test-utility" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-binding" id="InverseFunctions.test_inverse" href="#InverseFunctions.test_inverse"><code>InverseFunctions.test_inverse</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">InverseFunctions.test_inverse(f, x; compare=isapprox, kwargs...)</code></pre><p>Test if <a href="#InverseFunctions.inverse"><code>inverse(f)</code></a> is implemented correctly.</p><p>The function tests (as a <code>Test.@testset</code>) if</p><ul><li><code>compare(inverse(f)(f(x)), x) == true</code> and</li><li><code>compare(inverse(inverse(f))(x), f(x)) == true</code>.</li></ul><p><code>kwargs...</code> are forwarded to <code>compare</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/InverseFunctions.jl/blob/d397fef770dd28a78dd0df0d09e94c30b321f4db/src/test.jl#L1-L12">source</a></section></article><h2 id="Additional-functionality"><a class="docs-heading-anchor" href="#Additional-functionality">Additional functionality</a><a id="Additional-functionality-1"></a><a class="docs-heading-anchor-permalink" href="#Additional-functionality" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-binding" id="InverseFunctions.square" href="#InverseFunctions.square"><code>InverseFunctions.square</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">square(x::Real)</code></pre><p>Inverse of <code>sqrt(x)</code> for non-negative <code>x</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/InverseFunctions.jl/blob/d397fef770dd28a78dd0df0d09e94c30b321f4db/src/functions.jl#L3-L7">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="InverseFunctions.FunctionWithInverse" href="#InverseFunctions.FunctionWithInverse"><code>InverseFunctions.FunctionWithInverse</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct FunctionWithInverse{F,InvF} &lt;: Function</code></pre><p>A function with an inverse.</p><p>Do not construct directly, use <a href="#InverseFunctions.setinverse"><code>setinverse(f, invf)</code></a> instead.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/InverseFunctions.jl/blob/d397fef770dd28a78dd0df0d09e94c30b321f4db/src/setinverse.jl#L4-L10">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../LICENSE/">LICENSE »</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="Tuesday 1 August 2023 12:37">Tuesday 1 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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