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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>Inverting Integrals · DataInterpolations.jl</title><meta name="title" content="Inverting Integrals · DataInterpolations.jl"/><meta property="og:title" content="Inverting Integrals · DataInterpolations.jl"/><meta property="twitter:title" content="Inverting Integrals · DataInterpolations.jl"/><meta name="description" content="Documentation for DataInterpolations.jl."/><meta property="og:description" content="Documentation for DataInterpolations.jl."/><meta property="twitter:description" content="Documentation for DataInterpolations.jl."/><meta property="og:url" content="https://docs.sciml.ai/DataInterpolations/stable/inverting_integrals/"/><meta property="twitter:url" content="https://docs.sciml.ai/DataInterpolations/stable/inverting_integrals/"/><link rel="canonical" href="https://docs.sciml.ai/DataInterpolations/stable/inverting_integrals/"/><script data-outdated-warner src="../assets/warner.js"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><link href="../assets/favicon.ico" rel="icon" type="image/x-icon"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img src="../assets/logo.png" alt="DataInterpolations.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">DataInterpolations.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="../">DataInterpolations.jl</a></li><li><a class="tocitem" href="../methods/">Methods</a></li><li><a class="tocitem" href="../interface/">Interface</a></li><li><a class="tocitem" href="../manual/">Manual</a></li><li class="is-active"><a class="tocitem" href>Inverting Integrals</a><ul class="internal"><li><a class="tocitem" href="#Docstrings"><span>Docstrings</span></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 class="is-active"><a href>Inverting Integrals</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Inverting Integrals</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/SciML/DataInterpolations.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/SciML/DataInterpolations.jl/blob/master/docs/src/inverting_integrals.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="Inverting-integrals"><a class="docs-heading-anchor" href="#Inverting-integrals">Inverting integrals</a><a id="Inverting-integrals-1"></a><a class="docs-heading-anchor-permalink" href="#Inverting-integrals" title="Permalink"></a></h1><p>Solving implicit integral problems of the form:</p><p class="math-container">\[\begin{equation} | ||
| \text{find $t$ such that } \int_{t_1}^t f(\tau)\text{d}\tau = V \ge 0 | ||
| \end{equation}\]</p><p>is supported for interpolations <span>$f$</span> that are strictly positive and of one of these types:</p><ul><li><code>ConstantInterpolation</code></li><li><code>LinearInterpolation</code></li></ul><p>This is done by creating an 'integral inverse' interpolation object which can efficiently compute <span>$t$</span> for a given value of <span>$V$</span>, see the example below.</p><pre><code class="language-julia hljs">using DataInterpolations | ||
| using Plots | ||
|
|
||
| # Create LinearInterpolation object from the | ||
| u = sqrt.(1:25) + (2.0 * rand(25) .- 1.0) / 3 | ||
| t = cumsum(rand(25)) | ||
| A = LinearInterpolation(u, t) | ||
|
|
||
| # Create LinearInterpolationIntInv object | ||
| # from the LinearInterpolation object | ||
| A_intinv = DataInterpolations.invert_integral(A) | ||
|
|
||
| # Get the t values up to and including the | ||
| # solution to the integral problem | ||
| V = 25.0 | ||
| t_ = A_intinv(V) | ||
| ts = t[t .<= t_] | ||
| push!(ts, t_) | ||
|
|
||
| # Plot results | ||
| plot(A; label = "Linear Interpolation") | ||
| plot!(ts, A.(ts), fillrange = 0.0, fillalpha = 0.75, | ||
| fc = :blues, lw = 0, label = "Area of $V")</code></pre><img src="4bea52d7.svg" alt="Example block output"/><h2 id="Docstrings"><a class="docs-heading-anchor" href="#Docstrings">Docstrings</a><a id="Docstrings-1"></a><a class="docs-heading-anchor-permalink" href="#Docstrings" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="DataInterpolations.invert_integral" href="#DataInterpolations.invert_integral"><code>DataInterpolations.invert_integral</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">invert_integral(A::AbstractInterpolation)::AbstractIntegralInverseInterpolation</code></pre><p>Creates the inverted integral interpolation object from the given interpolation. Conditions:</p><ul><li>The range of <code>A</code> must be strictly positive</li><li>There must be an ordering defined on the data type of <code>A.u</code></li><li>This is currently only supported for ConstantInterpolation and LinearInterpolation</li></ul><p><strong>Arguments</strong></p><ul><li><code>A</code>: interpolation object satisfying the above requirements</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/DataInterpolations.jl/blob/152bcbfb1a06f415ce00eff1ac24f28f4fad40c0/src/integral_inverses.jl#L3-L15">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="DataInterpolations.ConstantInterpolationIntInv" href="#DataInterpolations.ConstantInterpolationIntInv"><code>DataInterpolations.ConstantInterpolationIntInv</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">some stuff</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/DataInterpolations.jl/blob/152bcbfb1a06f415ce00eff1ac24f28f4fad40c0/src/integral_inverses.jl#L58-L60">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="DataInterpolations.LinearInterpolationIntInv" href="#DataInterpolations.LinearInterpolationIntInv"><code>DataInterpolations.LinearInterpolationIntInv</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">some stuff</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/SciML/DataInterpolations.jl/blob/152bcbfb1a06f415ce00eff1ac24f28f4fad40c0/src/integral_inverses.jl#L24-L26">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../manual/">« Manual</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 17:09">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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