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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>About · IntervalMatrices.jl</title><script data-outdated-warner src="../assets/warner.js"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/0.045/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><link href="../assets/aligned.css" rel="stylesheet" type="text/css"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img src="../assets/logo.png" alt="IntervalMatrices.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">IntervalMatrices.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><span class="tocitem">Library</span><ul><li><a class="tocitem" href="../lib/types/">Types</a></li><li><a class="tocitem" href="../lib/methods/">Methods</a></li></ul></li><li class="is-active"><a class="tocitem" href>About</a><ul class="internal"><li><a class="tocitem" href="#Contributing"><span>Contributing</span></a></li><li><a class="tocitem" href="#Credits"><span>Credits</span></a></li></ul></li><li><a class="tocitem" href="../references/">References</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>About</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>About</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/JuliaReach/IntervalMatrices.jl/blob/master/docs/src/about.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="About"><a class="docs-heading-anchor" href="#About">About</a><a id="About-1"></a><a class="docs-heading-anchor-permalink" href="#About" title="Permalink"></a></h1><p>This page contains some general information about this project, and recommendations about contributing.</p><ul><li><a href="#About">About</a></li><li class="no-marker"><ul><li><a href="#Contributing">Contributing</a></li><li><a href="#Credits">Credits</a></li></ul></li></ul><h2 id="Contributing"><a class="docs-heading-anchor" href="#Contributing">Contributing</a><a id="Contributing-1"></a><a class="docs-heading-anchor-permalink" href="#Contributing" title="Permalink"></a></h2><p>If you like this package, consider contributing! You can send bug reports (or fix them and send your code), add examples to the documentation or propose new features.</p><p>Below we detail some of the guidelines that should be followed when contributing to this package. Further information can be found in the <a href="https://juliareach.github.io/JuliaReachDevDocs/latest/"><code>JuliaReachDevDocs</code> site</a>.</p><h3 id="Branches"><a class="docs-heading-anchor" href="#Branches">Branches</a><a id="Branches-1"></a><a class="docs-heading-anchor-permalink" href="#Branches" title="Permalink"></a></h3><p>Each pull request (PR) should be pushed in a new branch with the name of the author followed by a descriptive name, e.g. <code>mforets/my_feature</code>. If the branch is associated to a previous discussion in one issue, we use the name of the issue for easier lookup, e.g. <code>mforets/7</code>.</p><h3 id="Unit-testing-and-continuous-integration-(CI)"><a class="docs-heading-anchor" href="#Unit-testing-and-continuous-integration-(CI)">Unit testing and continuous integration (CI)</a><a id="Unit-testing-and-continuous-integration-(CI)-1"></a><a class="docs-heading-anchor-permalink" href="#Unit-testing-and-continuous-integration-(CI)" title="Permalink"></a></h3><p>This project is synchronized with GitHub Actions such that each PR gets tested before merging (and the build is automatically triggered after each new commit). For the maintainability of this project, it is important to make all unit tests pass.</p><p>To run the unit tests locally, you can do:</p><pre><code class="language-julia hljs">julia&gt; using Pkg

julia&gt; Pkg.test(&quot;IntervalMatrices&quot;)</code></pre><p>We also advise adding new unit tests when adding new features to ensure long-term support of your contributions.</p><h3 id="Contributing-to-the-documentation"><a class="docs-heading-anchor" href="#Contributing-to-the-documentation">Contributing to the documentation</a><a id="Contributing-to-the-documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Contributing-to-the-documentation" title="Permalink"></a></h3><p>This documentation is written in Markdown, and it relies on <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> to produce the HTML layout. To build the docs, run <code>make.jl</code>:</p><pre><code class="language-bash hljs">$ julia --color=yes docs/make.jl</code></pre><h2 id="Credits"><a class="docs-heading-anchor" href="#Credits">Credits</a><a id="Credits-1"></a><a class="docs-heading-anchor-permalink" href="#Credits" title="Permalink"></a></h2><p>These persons have contributed to <code>IntervalMatrices.jl</code> (in alphabetic order):</p><ul><li><a href="https://lucaferranti.github.io/">Luca Ferranti</a></li><li><a href="http://github.com/mforets">Marcelo Forets</a></li><li><a href="https://www.christianschilling.net/">Christian Schilling</a></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../lib/methods/">« Methods</a><a class="docs-footer-nextpage" href="../references/">References »</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 0.27.24 on <span class="colophon-date" title="Sunday 11 June 2023 09:28">Sunday 11 June 2023</span>. Using Julia version 1.9.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
julia&gt; Pkg.test(&quot;IntervalMatrices&quot;)</code></pre><p>We also advise adding new unit tests when adding new features to ensure long-term support of your contributions.</p><h3 id="Contributing-to-the-documentation"><a class="docs-heading-anchor" href="#Contributing-to-the-documentation">Contributing to the documentation</a><a id="Contributing-to-the-documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Contributing-to-the-documentation" title="Permalink"></a></h3><p>This documentation is written in Markdown, and it relies on <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> to produce the HTML layout. To build the docs, run <code>make.jl</code>:</p><pre><code class="language-bash hljs">$ julia --color=yes docs/make.jl</code></pre><h2 id="Credits"><a class="docs-heading-anchor" href="#Credits">Credits</a><a id="Credits-1"></a><a class="docs-heading-anchor-permalink" href="#Credits" title="Permalink"></a></h2><p>These persons have contributed to <code>IntervalMatrices.jl</code> (in alphabetic order):</p><ul><li><a href="https://lucaferranti.github.io/">Luca Ferranti</a></li><li><a href="http://github.com/mforets">Marcelo Forets</a></li><li><a href="https://www.christianschilling.net/">Christian Schilling</a></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../lib/methods/">« Methods</a><a class="docs-footer-nextpage" href="../references/">References »</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 0.27.24 on <span class="colophon-date" title="Saturday 24 June 2023 05:45">Saturday 24 June 2023</span>. Using Julia version 1.9.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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[-4.25, 3] [-2.25, 9]</code></pre><p>However, that result is not tight. The computation can be performed exactly via single-use expressions implemented in this library:</p><pre><code class="language-julia-repl hljs">julia&gt; quadratic_expansion(A, 1.0, 0.5)
2×2 IntervalMatrix{Float64, Interval{Float64}, Matrix{Interval{Float64}}}:
[1, 4.5] [-2, 1]
[-3, 1.5] [1, 7]</code></pre><p>We now obtain an interval matrix that is strictly included in the one obtained from the naive multiplication.</p><p>An overapproximation and an underapproximation method at a given order for <span>$e^{At}$</span>, where <span>$A$</span> is an interval matrix, are also available. See the <a href="lib/methods/#Methods">Methods</a> section for details.</p><h2 id="Library-Outline"><a class="docs-heading-anchor" href="#Library-Outline">Library Outline</a><a id="Library-Outline-1"></a><a class="docs-heading-anchor-permalink" href="#Library-Outline" title="Permalink"></a></h2><p>Explore the types and methods defined in this library by following the links below, or use the search bar in the left to look for a specific keyword in the documentation.</p><ul><li><a href="lib/types/#Types">Types</a></li><li class="no-marker"><ul><li><a href="lib/types/#Abstract-interval-operators">Abstract interval operators</a></li><li><a href="lib/types/#Interval-matrix">Interval matrix</a></li><li><a href="lib/types/#Interval-matrix-power-wrapper">Interval-matrix-power wrapper</a></li><li><a href="lib/types/#Affine-interval-matrix">Affine interval matrix</a></li></ul></li><li><a href="lib/methods/#Methods">Methods</a></li><li class="no-marker"><ul><li><a href="lib/methods/#Common-functions">Common functions</a></li><li><a href="lib/methods/#Arithmetic">Arithmetic</a></li><li><a href="lib/methods/#Matrix-power">Matrix power</a></li><li><a href="lib/methods/#Matrix-exponential">Matrix exponential</a></li><li><a href="lib/methods/#Finite-expansions">Finite expansions</a></li><li><a href="lib/methods/#Correction-terms">Correction terms</a></li><li><a href="lib/methods/#Norms">Norms</a></li></ul></li></ul><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>Althoff, Matthias, Olaf Stursberg, and Martin Buss. <em>Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization.</em> 2008 47th IEEE Conference on Decision and Control. IEEE, 2008.</li><li class="footnote" id="footnote-2"><a class="tag is-link" href="#citeref-2">2</a>Kosheleva, Olga, et al. <em>Computing the cube of an interval matrix is NP-hard.</em> Proceedings of the 2005 ACM symposium on Applied computing. ACM, 2005.</li><li class="footnote" id="footnote-3"><a class="tag is-link" href="#citeref-3">3</a>Althoff, Matthias, Bruce H. Krogh, and Olaf Stursberg. <em>Analyzing reachability of linear dynamic systems with parametric uncertainties.</em> Modeling, Design, and Simulation of Systems with Uncertainties. Springer, Berlin, Heidelberg, 2011. 69-94.</li><li class="footnote" id="footnote-4"><a class="tag is-link" href="#citeref-4">4</a>Liou, M. L. <em>A novel method of evaluating transient response.</em> Proceedings of the IEEE 54.1 (1966): 20-23.</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="lib/types/">Types »</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 0.27.24 on <span class="colophon-date" title="Sunday 11 June 2023 09:28">Sunday 11 June 2023</span>. Using Julia version 1.9.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
[-3, 1.5] [1, 7]</code></pre><p>We now obtain an interval matrix that is strictly included in the one obtained from the naive multiplication.</p><p>An overapproximation and an underapproximation method at a given order for <span>$e^{At}$</span>, where <span>$A$</span> is an interval matrix, are also available. See the <a href="lib/methods/#Methods">Methods</a> section for details.</p><h2 id="Library-Outline"><a class="docs-heading-anchor" href="#Library-Outline">Library Outline</a><a id="Library-Outline-1"></a><a class="docs-heading-anchor-permalink" href="#Library-Outline" title="Permalink"></a></h2><p>Explore the types and methods defined in this library by following the links below, or use the search bar in the left to look for a specific keyword in the documentation.</p><ul><li><a href="lib/types/#Types">Types</a></li><li class="no-marker"><ul><li><a href="lib/types/#Abstract-interval-operators">Abstract interval operators</a></li><li><a href="lib/types/#Interval-matrix">Interval matrix</a></li><li><a href="lib/types/#Interval-matrix-power-wrapper">Interval-matrix-power wrapper</a></li><li><a href="lib/types/#Affine-interval-matrix">Affine interval matrix</a></li></ul></li><li><a href="lib/methods/#Methods">Methods</a></li><li class="no-marker"><ul><li><a href="lib/methods/#Common-functions">Common functions</a></li><li><a href="lib/methods/#Arithmetic">Arithmetic</a></li><li><a href="lib/methods/#Matrix-power">Matrix power</a></li><li><a href="lib/methods/#Matrix-exponential">Matrix exponential</a></li><li><a href="lib/methods/#Finite-expansions">Finite expansions</a></li><li><a href="lib/methods/#Correction-terms">Correction terms</a></li><li><a href="lib/methods/#Norms">Norms</a></li></ul></li></ul><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>Althoff, Matthias, Olaf Stursberg, and Martin Buss. <em>Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization.</em> 2008 47th IEEE Conference on Decision and Control. IEEE, 2008.</li><li class="footnote" id="footnote-2"><a class="tag is-link" href="#citeref-2">2</a>Kosheleva, Olga, et al. <em>Computing the cube of an interval matrix is NP-hard.</em> Proceedings of the 2005 ACM symposium on Applied computing. ACM, 2005.</li><li class="footnote" id="footnote-3"><a class="tag is-link" href="#citeref-3">3</a>Althoff, Matthias, Bruce H. Krogh, and Olaf Stursberg. <em>Analyzing reachability of linear dynamic systems with parametric uncertainties.</em> Modeling, Design, and Simulation of Systems with Uncertainties. Springer, Berlin, Heidelberg, 2011. 69-94.</li><li class="footnote" id="footnote-4"><a class="tag is-link" href="#citeref-4">4</a>Liou, M. L. <em>A novel method of evaluating transient response.</em> Proceedings of the IEEE 54.1 (1966): 20-23.</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="lib/types/">Types »</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 0.27.24 on <span class="colophon-date" title="Saturday 24 June 2023 05:46">Saturday 24 June 2023</span>. Using Julia version 1.9.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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