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<h1 class="title">Sharktooth Functions in 1-dimension</h1>
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<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np </span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> max_conv_operator(samples, f_samples, <span class="bu">input</span>, L):</span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> np.<span class="bu">max</span>(f_samples <span class="op">-</span> L <span class="op">*</span> np.<span class="bu">abs</span>(<span class="bu">input</span> <span class="op">-</span> samples))</span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> sharktooth_function(function, x_start, x_stop, number_of_sharkteeth, L, plot_arg):</span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a> samples <span class="op">=</span> np.random.uniform(x_start,x_stop, number_of_sharkteeth)</span>
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a> f_samples <span class="op">=</span> function(samples)</span>
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a> x <span class="op">=</span> np.linspace(x_start, x_stop, <span class="dv">10000</span>)</span>
<span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a> approximate_y <span class="op">=</span> []</span>
<span id="cb2-9"><a href="#cb2-9" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(<span class="bu">len</span>(x)):</span>
<span id="cb2-10"><a href="#cb2-10" aria-hidden="true" tabindex="-1"></a> approximate_y.append(max_conv_operator(samples, f_samples, x[i], L))</span>
<span id="cb2-11"><a href="#cb2-11" aria-hidden="true" tabindex="-1"></a> error <span class="op">=</span> np.<span class="bu">max</span>(np.<span class="bu">abs</span>(f(x) <span class="op">-</span> approximate_y))</span>
<span id="cb2-12"><a href="#cb2-12" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> plot_arg <span class="op">==</span> <span class="dv">1</span>:</span>
<span id="cb2-13"><a href="#cb2-13" aria-hidden="true" tabindex="-1"></a> plt.plot(x,approximate_y)</span>
<span id="cb2-14"><a href="#cb2-14" aria-hidden="true" tabindex="-1"></a> <span class="cf">else</span>:</span>
<span id="cb2-15"><a href="#cb2-15" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> error</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> f(x):</span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> x<span class="op">/</span>(x<span class="op">+</span><span class="dv">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>L <span class="op">=</span> [<span class="dv">1</span>,<span class="dv">4</span>,<span class="dv">8</span>,<span class="dv">16</span>]</span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>plt.figure(figsize<span class="op">=</span>(<span class="dv">17</span>,<span class="dv">5</span>))</span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> L <span class="kw">in</span> L:</span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a> errors <span class="op">=</span> []</span>
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(<span class="dv">1</span>,<span class="dv">50</span>):</span>
<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a> errors.append(sharktooth_function(f,<span class="dv">0</span>,<span class="dv">1</span>,i,L,<span class="dv">0</span>))</span>
<span id="cb4-7"><a href="#cb4-7" aria-hidden="true" tabindex="-1"></a> <span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</span>
<span id="cb4-8"><a href="#cb4-8" aria-hidden="true" tabindex="-1"></a> plt.plot(errors, label <span class="op">=</span>L)</span>
<span id="cb4-9"><a href="#cb4-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-10"><a href="#cb4-10" aria-hidden="true" tabindex="-1"></a>plt.xlabel(<span class="st">"Number of teeth"</span>)</span>
<span id="cb4-11"><a href="#cb4-11" aria-hidden="true" tabindex="-1"></a>plt.ylabel(<span class="vs">r"$\sup_{x\in x_i} | f(x)- \mathfrak</span><span class="sc">{R}</span><span class="vs">_{\mathfrak</span><span class="sc">{r}</span><span class="vs">} ( \mathsf</span><span class="sc">{P}</span><span class="vs">)|$"</span>)</span>
<span id="cb4-12"><a href="#cb4-12" aria-hidden="true" tabindex="-1"></a>plt.legend()</span>
<span id="cb4-13"><a href="#cb4-13" aria-hidden="true" tabindex="-1"></a>plt.title(<span class="st">"Sup of deviance from f(x) as the number of teeth increase and as we get closer to L"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>Text(0.5, 1.0, 'Sup of deviance from f(x) as the number of teeth increase and as we get closer to L')</code></pre>
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