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dev/getting_started_with_julia/index.html
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@ -14,4 +14,4 @@ julia&gt; DataFrame(a=[1,2],b=[3,4])</code></pre><p>You should get an error or a
BenchmarkTools = &quot;6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf&quot; BenchmarkTools = &quot;6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf&quot;
DataFrames = &quot;a93c6f00-e57d-5684-b7b6-d8193f3e46c0&quot; DataFrames = &quot;a93c6f00-e57d-5684-b7b6-d8193f3e46c0&quot;
MPI = &quot;da04e1cc-30fd-572f-bb4f-1f8673147195&quot;</code></pre><p>Copy the contents of previous code block into a file called <code>Project.toml</code> and place it in an empty folder named <code>newproject</code>. It is important that the file is named <code>Project.toml</code>. You can create a new folder from the REPL with</p><pre><code class="language-julia hljs">julia&gt; mkdir(&quot;newproject&quot;)</code></pre><p>To install all the packages registered in this file you need to activate the folder containing your <code>Project.toml</code> file</p><pre><code class="language-julia hljs">(@v1.8) pkg&gt; activate newproject</code></pre><p>and then <em>instantiating</em> it</p><pre><code class="language-julia hljs">(newproject) pkg&gt; instantiate</code></pre><p>The instantiate command will download and install all listed packages and their dependencies in just one click.</p><h3 id="Getting-help-in-package-mode"><a class="docs-heading-anchor" href="#Getting-help-in-package-mode">Getting help in package mode</a><a id="Getting-help-in-package-mode-1"></a><a class="docs-heading-anchor-permalink" href="#Getting-help-in-package-mode" title="Permalink"></a></h3><p>You can get help about a particular package operator by writing <code>help</code> in front of it</p><pre><code class="language-julia hljs">(@v1.8) pkg&gt; help activate</code></pre><p>You can get an overview of all package commands by typing <code>help</code> alone</p><pre><code class="language-julia hljs">(@v1.8) pkg&gt; help</code></pre><h3 id="Package-operations-in-Julia-code"><a class="docs-heading-anchor" href="#Package-operations-in-Julia-code">Package operations in Julia code</a><a id="Package-operations-in-Julia-code-1"></a><a class="docs-heading-anchor-permalink" href="#Package-operations-in-Julia-code" title="Permalink"></a></h3><p>In some situations it is required to use package commands in Julia code, e.g., to automatize installation and deployment of Julia applications. This can be done using the <code>Pkg</code> package. For instance</p><pre><code class="language-julia hljs">julia&gt; using Pkg MPI = &quot;da04e1cc-30fd-572f-bb4f-1f8673147195&quot;</code></pre><p>Copy the contents of previous code block into a file called <code>Project.toml</code> and place it in an empty folder named <code>newproject</code>. It is important that the file is named <code>Project.toml</code>. You can create a new folder from the REPL with</p><pre><code class="language-julia hljs">julia&gt; mkdir(&quot;newproject&quot;)</code></pre><p>To install all the packages registered in this file you need to activate the folder containing your <code>Project.toml</code> file</p><pre><code class="language-julia hljs">(@v1.8) pkg&gt; activate newproject</code></pre><p>and then <em>instantiating</em> it</p><pre><code class="language-julia hljs">(newproject) pkg&gt; instantiate</code></pre><p>The instantiate command will download and install all listed packages and their dependencies in just one click.</p><h3 id="Getting-help-in-package-mode"><a class="docs-heading-anchor" href="#Getting-help-in-package-mode">Getting help in package mode</a><a id="Getting-help-in-package-mode-1"></a><a class="docs-heading-anchor-permalink" href="#Getting-help-in-package-mode" title="Permalink"></a></h3><p>You can get help about a particular package operator by writing <code>help</code> in front of it</p><pre><code class="language-julia hljs">(@v1.8) pkg&gt; help activate</code></pre><p>You can get an overview of all package commands by typing <code>help</code> alone</p><pre><code class="language-julia hljs">(@v1.8) pkg&gt; help</code></pre><h3 id="Package-operations-in-Julia-code"><a class="docs-heading-anchor" href="#Package-operations-in-Julia-code">Package operations in Julia code</a><a id="Package-operations-in-Julia-code-1"></a><a class="docs-heading-anchor-permalink" href="#Package-operations-in-Julia-code" title="Permalink"></a></h3><p>In some situations it is required to use package commands in Julia code, e.g., to automatize installation and deployment of Julia applications. This can be done using the <code>Pkg</code> package. For instance</p><pre><code class="language-julia hljs">julia&gt; using Pkg
julia&gt; Pkg.status()</code></pre><p>is equivalent to calling <code>status</code> in package mode.</p><pre><code class="language-julia hljs">(@v1.8) pkg&gt; status</code></pre><h2 id="Conclusion"><a class="docs-heading-anchor" href="#Conclusion">Conclusion</a><a id="Conclusion-1"></a><a class="docs-heading-anchor-permalink" href="#Conclusion" title="Permalink"></a></h2><p>We have learned the basics of how to work with Julia. If you want to further dig into the topics we have covered here, you can take a look at the following links:</p><ul><li><a href="https://docs.julialang.org/en/v1/manual/getting-started/">Julia Manual</a></li><li><a href="https://pkgdocs.julialang.org/v1/getting-started/">Package manager</a></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../notebooks/julia_basics/">Julia Basics »</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 24 August 2023 14:59">Thursday 24 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> julia&gt; Pkg.status()</code></pre><p>is equivalent to calling <code>status</code> in package mode.</p><pre><code class="language-julia hljs">(@v1.8) pkg&gt; status</code></pre><h2 id="Conclusion"><a class="docs-heading-anchor" href="#Conclusion">Conclusion</a><a id="Conclusion-1"></a><a class="docs-heading-anchor-permalink" href="#Conclusion" title="Permalink"></a></h2><p>We have learned the basics of how to work with Julia. If you want to further dig into the topics we have covered here, you can take a look at the following links:</p><ul><li><a href="https://docs.julialang.org/en/v1/manual/getting-started/">Julia Manual</a></li><li><a href="https://pkgdocs.julialang.org/v1/getting-started/">Package manager</a></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../notebooks/julia_basics/">Julia Basics »</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="Friday 25 August 2023 13:20">Friday 25 August 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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@ -7438,66 +7438,57 @@ a.anchor-link {
<!-- End of mermaid configuration --></head> <!-- End of mermaid configuration --></head>
<body class="jp-Notebook" data-jp-theme-light="true" data-jp-theme-name="JupyterLab Light"> <body class="jp-Notebook" data-jp-theme-light="true" data-jp-theme-name="JupyterLab Light">
<main> <main>
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<h1 id="Solving-Linear-Equations">Solving Linear Equations<a class="anchor-link" href="#Solving-Linear-Equations"></a></h1><h2 id="Serial-Algorithm">Serial Algorithm<a class="anchor-link" href="#Serial-Algorithm"></a></h2><p>First, we construct a linear equations system $Ax=b$.</p> <h1 id="Solving-Linear-Equations">Solving Linear Equations<a class="anchor-link" href="#Solving-Linear-Equations"></a></h1><h2 id="Serial-Algorithm">Serial Algorithm<a class="anchor-link" href="#Serial-Algorithm"></a></h2><p>To demonstrate the algorithm, we will consider a simple system of linear equations $Ax = b$:</p>
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<div class="jp-InputPrompt jp-InputArea-prompt">In [ ]:</div> <div class="jp-InputPrompt jp-InputArea-prompt">In [66]:</div>
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<div class="highlight hl-julia"><pre><span></span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">5</span> <div class="highlight hl-julia"><pre><span></span><span class="n">A</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">1.0</span><span class="w"> </span><span class="mf">4.0</span><span class="w"> </span><span class="mf">5.0</span><span class="w"> </span><span class="mf">8.0</span><span class="w"> </span><span class="mf">1.0</span><span class="p">;</span><span class="w"> </span>
<span class="n">A</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rand</span><span class="p">(</span><span class="o">-</span><span class="mf">10.0</span><span class="o">:</span><span class="mf">10.0</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">))</span> <span class="w"> </span><span class="mf">2.0</span><span class="w"> </span><span class="o">-</span><span class="mf">1.0</span><span class="w"> </span><span class="mf">4.0</span><span class="w"> </span><span class="mf">3.0</span><span class="w"> </span><span class="mf">0.0</span><span class="p">;</span><span class="w"> </span>
</pre></div> <span class="w"> </span><span class="mf">7.0</span><span class="w"> </span><span class="mf">6.0</span><span class="w"> </span><span class="mf">3.0</span><span class="w"> </span><span class="o">-</span><span class="mf">4.0</span><span class="w"> </span><span class="mf">5.0</span><span class="p">;</span><span class="w"> </span>
</div> <span class="w"> </span><span class="o">-</span><span class="mf">3.0</span><span class="w"> </span><span class="mf">4.0</span><span class="w"> </span><span class="mf">2.0</span><span class="w"> </span><span class="mf">2.0</span><span class="w"> </span><span class="mf">2.0</span><span class="p">;</span><span class="w"> </span>
</div> <span class="w"> </span><span class="mf">0.0</span><span class="w"> </span><span class="o">-</span><span class="mf">4.0</span><span class="w"> </span><span class="mf">2.0</span><span class="w"> </span><span class="mf">1.0</span><span class="w"> </span><span class="mf">2.0</span><span class="p">]</span>
</div>
</div> <span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">61.0</span><span class="p">;</span><span class="w"> </span><span class="mf">24.0</span><span class="p">;</span><span class="w"> </span><span class="mf">37.0</span><span class="p">;</span><span class="w"> </span><span class="mf">29.0</span><span class="p">;</span><span class="w"> </span><span class="mf">12.0</span><span class="p">];</span>
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<div class="highlight hl-julia"><pre><span></span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rand</span><span class="p">(</span><span class="o">-</span><span class="mf">10.0</span><span class="o">:</span><span class="mf">10.0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
<span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">A</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">x</span>
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<p>The code in the following cell converts the problem $Ax=b$ to the upper triangular equation system $Ux=y$.</p> <p>The code in the following cell converts the general problem $Ax=b$ to the upper triangular equation system $Ux=y$. Note that this function assumes that the pivots are all nonzero. This function will be erroneos if any of the diagonal entries are zero!</p>
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<div class="jp-InputPrompt jp-InputArea-prompt">In [ ]:</div> <div class="jp-InputPrompt jp-InputArea-prompt">In [67]:</div>
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<div class="highlight hl-julia"><pre><span></span><span class="k">function</span><span class="w"> </span><span class="n">convert_to_upper_triangular</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">b</span><span class="p">)</span> <div class="highlight hl-julia"><pre><span></span><span class="k">function</span><span class="w"> </span><span class="n">convert_to_upper_triangular!</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">b</span><span class="p">)</span>
<span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="w"> </span><span class="c"># Upper Triangularization: convert Ax=b to Ux=y</span> <span class="w"> </span><span class="c"># Upper Triangularization: convert Ax=b to Ux=y</span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">:</span><span class="n">n</span> <span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">:</span><span class="n">n</span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="o">:</span><span class="n">n</span> <span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="o">:</span><span class="n">n</span>
@ -7522,26 +7513,38 @@ a.anchor-link {
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<pre>convert_to_upper_triangular! (generic function with 1 method)</pre>
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<p>The function in the following cell solves the upper triangular equation system using backwards substitution.</p> <p>The function in the following cell solves the upper triangular equation system using backwards substitution. Note that the function alters the input values.</p>
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<div class="jp-InputPrompt jp-InputArea-prompt">In [ ]:</div> <div class="jp-InputPrompt jp-InputArea-prompt">In [68]:</div>
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<div class="highlight hl-julia"><pre><span></span><span class="k">function</span><span class="w"> </span><span class="n">solve_upper_triangular</span><span class="p">(</span><span class="n">U</span><span class="p">,</span><span class="n">y</span><span class="p">)</span> <div class="highlight hl-julia"><pre><span></span><span class="k">function</span><span class="w"> </span><span class="n">solve_upper_triangular!</span><span class="p">(</span><span class="n">U</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
<span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">U</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span> <span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">U</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">step</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">reverse</span><span class="p">(</span><span class="mi">1</span><span class="o">:</span><span class="n">n</span><span class="p">)</span> <span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">step</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">reverse</span><span class="p">(</span><span class="mi">1</span><span class="o">:</span><span class="n">n</span><span class="p">)</span>
<span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">U</span><span class="p">[</span><span class="n">step</span><span class="p">,</span><span class="n">step</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span> <span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">U</span><span class="p">[</span><span class="n">step</span><span class="p">,</span><span class="n">step</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
@ -7565,23 +7568,52 @@ a.anchor-link {
</div> </div>
</div> </div>
</div> </div>
</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell jp-mod-noOutputs" id="cell-id=2c24d85e"> <div class="jp-Cell-outputWrapper">
<div class="jp-Collapser jp-OutputCollapser jp-Cell-outputCollapser">
</div>
<div class="jp-OutputArea jp-Cell-outputArea">
<div class="jp-OutputArea-child jp-OutputArea-executeResult">
<div class="jp-OutputPrompt jp-OutputArea-prompt">Out[68]:</div>
<div class="jp-RenderedText jp-OutputArea-output jp-OutputArea-executeResult" data-mime-type="text/plain" tabindex="0">
<pre>solve_upper_triangular! (generic function with 1 method)</pre>
</div>
</div>
</div>
</div>
</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell" id="cell-id=b92332f7">
<div class="jp-Cell-inputWrapper" tabindex="0"> <div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser"> <div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div> </div>
<div class="jp-InputArea jp-Cell-inputArea"> <div class="jp-InputArea jp-Cell-inputArea">
<div class="jp-InputPrompt jp-InputArea-prompt">In [ ]:</div> <div class="jp-InputPrompt jp-InputArea-prompt">In [69]:</div>
<div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline"> <div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline">
<div class="cm-editor cm-s-jupyter"> <div class="cm-editor cm-s-jupyter">
<div class="highlight hl-julia"><pre><span></span><span class="n">U</span><span class="p">,</span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">convert_to_upper_triangular</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">b</span><span class="p">)</span> <div class="highlight hl-julia"><pre><span></span><span class="n">U</span><span class="p">,</span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">convert_to_upper_triangular!</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">b</span><span class="p">)</span>
<span class="n">sol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">solve_upper_triangular</span><span class="p">(</span><span class="n">U</span><span class="p">,</span><span class="n">y</span><span class="p">)</span> <span class="n">sol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">solve_upper_triangular!</span><span class="p">(</span><span class="n">U</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
</pre></div> </pre></div>
</div> </div>
</div> </div>
</div> </div>
</div> </div>
<div class="jp-Cell-outputWrapper">
<div class="jp-Collapser jp-OutputCollapser jp-Cell-outputCollapser">
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<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=1c356b5a"> <div class="jp-OutputArea jp-Cell-outputArea">
<div class="jp-OutputArea-child jp-OutputArea-executeResult">
<div class="jp-OutputPrompt jp-OutputArea-prompt">Out[69]:</div>
<div class="jp-RenderedText jp-OutputArea-output jp-OutputArea-executeResult" data-mime-type="text/plain" tabindex="0">
<pre>5-element Vector{Float64}:
1.0000000000000009
1.999999999999999
2.9999999999999964
4.000000000000002
5.000000000000005</pre>
</div>
</div>
</div>
</div>
</div>
<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=962ec4a9">
<div class="jp-Cell-inputWrapper" tabindex="0"> <div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser"> <div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div> </div>
@ -7591,22 +7623,34 @@ a.anchor-link {
</div> </div>
</div> </div>
</div> </div>
</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell jp-mod-noOutputs" id="cell-id=5c71828d"> </div><div class="jp-Cell jp-CodeCell jp-Notebook-cell" id="cell-id=0e336c85">
<div class="jp-Cell-inputWrapper" tabindex="0"> <div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser"> <div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div> </div>
<div class="jp-InputArea jp-Cell-inputArea"> <div class="jp-InputArea jp-Cell-inputArea">
<div class="jp-InputPrompt jp-InputArea-prompt">In [ ]:</div> <div class="jp-InputPrompt jp-InputArea-prompt">In [70]:</div>
<div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline"> <div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline">
<div class="cm-editor cm-s-jupyter"> <div class="cm-editor cm-s-jupyter">
<div class="highlight hl-julia"><pre><span></span><span class="k">using</span><span class="w"> </span><span class="n">Test</span> <div class="highlight hl-julia"><pre><span></span><span class="k">using</span><span class="w"> </span><span class="n">Test</span>
<span class="nd">@test</span><span class="w"> </span><span class="n">sol</span><span class="w"> </span><span class="o">≈</span><span class="w"> </span><span class="n">x</span> <span class="nd">@test</span><span class="w"> </span><span class="n">sol</span><span class="w"> </span><span class="o">≈</span><span class="w"> </span><span class="p">[</span><span class="mf">1.0</span><span class="p">;</span><span class="w"> </span><span class="mf">2.0</span><span class="p">;</span><span class="w"> </span><span class="mf">3.0</span><span class="p">;</span><span class="w"> </span><span class="mf">4.0</span><span class="p">;</span><span class="w"> </span><span class="mf">5.0</span><span class="p">]</span>
</pre></div> </pre></div>
</div> </div>
</div> </div>
</div> </div>
</div> </div>
</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell jp-mod-noOutputs" id="cell-id=fe3c5374"> <div class="jp-Cell-outputWrapper">
<div class="jp-Collapser jp-OutputCollapser jp-Cell-outputCollapser">
</div>
<div class="jp-OutputArea jp-Cell-outputArea">
<div class="jp-OutputArea-child jp-OutputArea-executeResult">
<div class="jp-OutputPrompt jp-OutputArea-prompt">Out[70]:</div>
<div class="jp-RenderedText jp-OutputArea-output jp-OutputArea-executeResult" data-mime-type="text/plain" tabindex="0">
<pre><span class="ansi-green-intense-fg ansi-bold">Test Passed</span></pre>
</div>
</div>
</div>
</div>
</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell jp-mod-noOutputs" id="cell-id=28dff449">
<div class="jp-Cell-inputWrapper" tabindex="0"> <div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser"> <div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div> </div>

124
dev/notebooks/LEQ.ipynb
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@ -2,53 +2,59 @@
"cells": [ "cells": [
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "599913a3", "id": "9c32b051",
"metadata": {}, "metadata": {},
"source": [ "source": [
"# Solving Linear Equations\n", "# Solving Linear Equations\n",
"\n", "\n",
"## Serial Algorithm\n", "## Serial Algorithm\n",
"First, we construct a linear equations system $Ax=b$." "To demonstrate the algorithm, we will consider a simple system of linear equations $Ax = b$:"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 66,
"id": "fe3b5b02", "id": "2b369b73",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
"n = 5\n", "A = [1.0 4.0 5.0 8.0 1.0; \n",
"A = rand(-10.0:10.0, (n, n))" " 2.0 -1.0 4.0 3.0 0.0; \n",
] " 7.0 6.0 3.0 -4.0 5.0; \n",
}, " -3.0 4.0 2.0 2.0 2.0; \n",
{ " 0.0 -4.0 2.0 1.0 2.0]\n",
"cell_type": "code", "\n",
"execution_count": null, "b = [61.0; 24.0; 37.0; 29.0; 12.0];"
"id": "57c912fb",
"metadata": {},
"outputs": [],
"source": [
"x = rand(-10.0:10.0, n)\n",
"b = A * x"
] ]
}, },
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "4baa0681", "id": "53124eb8",
"metadata": {}, "metadata": {},
"source": [ "source": [
"The code in the following cell converts the problem $Ax=b$ to the upper triangular equation system $Ux=y$. " "The code in the following cell converts the general problem $Ax=b$ to the upper triangular equation system $Ux=y$. Note that this function assumes that the pivots are all nonzero. This function will be erroneos if any of the diagonal entries are zero!"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 67,
"id": "11d255e0", "id": "7a7b926a",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"convert_to_upper_triangular! (generic function with 1 method)"
]
},
"execution_count": 67,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"function convert_to_upper_triangular(A,b)\n", "function convert_to_upper_triangular!(A,b)\n",
" n = size(A,1)\n",
" # Upper Triangularization: convert Ax=b to Ux=y\n", " # Upper Triangularization: convert Ax=b to Ux=y\n",
" for k in 1:n\n", " for k in 1:n\n",
" for j in k+1:n\n", " for j in k+1:n\n",
@ -72,20 +78,31 @@
}, },
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "02c47593", "id": "78ef1849",
"metadata": {}, "metadata": {},
"source": [ "source": [
"The function in the following cell solves the upper triangular equation system using backwards substitution. " "The function in the following cell solves the upper triangular equation system using backwards substitution. Note that the function alters the input values. "
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 68,
"id": "3c19497c", "id": "5a134433",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"solve_upper_triangular! (generic function with 1 method)"
]
},
"execution_count": 68,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"function solve_upper_triangular(U,y)\n", "function solve_upper_triangular!(U,y)\n",
" n = size(U,1)\n", " n = size(U,1)\n",
" for step in reverse(1:n)\n", " for step in reverse(1:n)\n",
" if U[step,step] == 0\n", " if U[step,step] == 0\n",
@ -108,18 +125,34 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 69,
"id": "2c24d85e", "id": "b92332f7",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"5-element Vector{Float64}:\n",
" 1.0000000000000009\n",
" 1.999999999999999\n",
" 2.9999999999999964\n",
" 4.000000000000002\n",
" 5.000000000000005"
]
},
"execution_count": 69,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"U,y = convert_to_upper_triangular(A,b)\n", "U,y = convert_to_upper_triangular!(A,b)\n",
"sol = solve_upper_triangular(U,y)" "sol = solve_upper_triangular!(U,y)"
] ]
}, },
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "1c356b5a", "id": "962ec4a9",
"metadata": {}, "metadata": {},
"source": [ "source": [
"We can test if the obtained solution is correct using `@test`:" "We can test if the obtained solution is correct using `@test`:"
@ -127,19 +160,30 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 70,
"id": "5c71828d", "id": "0e336c85",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"\u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m"
]
},
"execution_count": 70,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"using Test\n", "using Test\n",
"@test sol ≈ x" "@test sol ≈ [1.0; 2.0; 3.0; 4.0; 5.0]"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": null,
"id": "fe3c5374", "id": "28dff449",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [] "source": []

2
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var myIframe = document.getElementById("notebook"); var myIframe = document.getElementById("notebook");
iFrameResize({log:true}, myIframe); iFrameResize({log:true}, myIframe);
}); });
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@ -18,4 +18,4 @@
var myIframe = document.getElementById("notebook"); var myIframe = document.getElementById("notebook");
iFrameResize({log:true}, myIframe); iFrameResize({log:true}, myIframe);
}); });
</script></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../jacobi_method/">« Jacobi</a><a class="docs-footer-nextpage" href="../solutions/">Solutions »</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 24 August 2023 14:59">Thursday 24 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> </script></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../jacobi_method/">« Jacobi</a><a class="docs-footer-nextpage" href="../solutions/">Solutions »</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="Friday 25 August 2023 13:20">Friday 25 August 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

2
dev/notebooks/jacobi_2D/index.html
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@ -18,4 +18,4 @@
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