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Documenter.jl
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<!-- End of mermaid configuration --></head>
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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>
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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>To demonstrate the algorithm, we will consider a simple system of linear equations $Ax = b$:</p>
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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>
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<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>
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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>
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<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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<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>
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<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>
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<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>
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<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>
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<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>
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<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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<p>The code in the following cell converts the problem $Ax=b$ to the upper triangular equation system $Ux=y$.</p>
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<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="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>
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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>
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<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>
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<span class="w"> </span><span class="c"># Upper Triangularization: convert Ax=b to Ux=y</span>
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<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>
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<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>
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@@ -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>
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<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="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>
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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>
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<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>
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<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>
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<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>
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@@ -7565,23 +7568,52 @@ a.anchor-link {
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<pre>solve_upper_triangular! (generic function with 1 method)</pre>
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<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>
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<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>
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<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>
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<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>
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<pre>5-element Vector{Float64}:
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1.0000000000000009
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1.999999999999999
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2.9999999999999964
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4.000000000000002
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5.000000000000005</pre>
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<div class="highlight hl-julia"><pre><span></span><span class="k">using</span><span class="w"> </span><span class="n">Test</span>
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<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>
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<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>
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|
||||
<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-Collapser jp-InputCollapser jp-Cell-inputCollapser">
|
||||
</div>
|
||||
|
||||
Reference in New Issue
Block a user