move geometry to geometry2d and add new geometry module (3d)

2019-12-21 12:51:34 +01:00
parent 9319d1bb06
commit 95a289be75
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+  <div class="section" id="module-pylib.numerical.integration">
+<span id="pylib-numerical-integration-module"></span><h1>pylib.numerical.integration module<a class="headerlink" href="#module-pylib.numerical.integration" title="Permalink to this headline">¶</a></h1>
+<p>Numerical integration, numerical quadrature.</p>
+<p>de: numerische Integration, numerische Quadratur.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Date</dt>
+<dd class="field-odd"><p>2015-10-15</p>
+</dd>
+</dl>
+<span class="target" id="module-integration"></span><dl class="function">
+<dt id="pylib.numerical.integration.trapez">
+<code class="sig-name descname">trapez</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">a=0</em>, <em class="sig-param">b=1</em>, <em class="sig-param">N=10</em>, <em class="sig-param">x=None</em>, <em class="sig-param">verbose=False</em>, <em class="sig-param">save_values=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/numerical/integration.html#trapez"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.numerical.integration.trapez" title="Permalink to this definition">¶</a></dt>
+<dd><p>Integration of <span class="math notranslate nohighlight">\(f(x)\)</span> using the trapezoidal rule
+(Simpson’s rule, Kepler’s rule).</p>
+<p>de: Trapezregel, Simpsonregel (Thomas Simpson), Keplersche
+Fassregel (Johannes Kepler)</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters</dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>f</strong> (<em>function</em><em> or </em><em>list</em>) – function to integrate.</p></li>
+<li><p><strong>a</strong> (<em>float</em>) – lower limit of integration (default = 0).</p></li>
+<li><p><strong>b</strong> (<em>float</em>) – upper limit of integration (default = 1).</p></li>
+<li><p><strong>N</strong> (<em>int</em>) – specify the number of subintervals.</p></li>
+<li><p><strong>x</strong> (<em>list</em>) – variable of integration, necessary if f is a list
+(default = None).</p></li>
+<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns</dt>
+<dd class="field-even"><p>the definite integral as approximated by trapezoidal
+rule.</p>
+</dd>
+<dt class="field-odd">Return type</dt>
+<dd class="field-odd"><p>float</p>
+</dd>
+</dl>
+<p>The trapezoidal rule approximates the integral by the area of a
+trapezoid with base h=b-a and sides equal to the values of the
+integrand at the two end points.</p>
+<div class="math notranslate nohighlight">
+\[f_n(x) = f(a)+\frac{f(b)-f(a)}{b-a}(x-a)\]</div>
+<div class="math notranslate nohighlight">
+\[\begin{split}I &amp;= \int\limits_a^b f(x) \,\mathrm{d}x \\
+I &amp;\approx \int\limits_a^b f_n(x) \,\mathrm{d}x \\
+  &amp;= \int\limits_a^b
+     \left( f(a)+\frac{f(b)-f(a)}{b-a}(x-a) \right)
+     \mathrm{d}x \\
+  &amp;= \left.\left( f(a)-a\frac{f(b)-f(a)}{b-a} \right)
+     x \right\vert_a^b +
+     \left. \frac{f(b)-f(a)}{b-a} \frac{x^2}{2}
+     \right\vert_a^b \\
+  &amp;= \frac{b-a}{2}\left[f(a)+f(b)\right]\end{split}\]</div>
+<p>The composite trapezium rule. If the interval is divided into n
+segments (not necessarily equal)</p>
+<div class="math notranslate nohighlight">
+\[a = x_0 \leq x_1 \leq x_2 \leq \ldots \leq x_n = b\]</div>
+<div class="math notranslate nohighlight">
+\[\begin{split}I &amp;\approx \sum\limits_{i=0}^{n-1} \frac{1}{2} (x_{i+1}-x_i)
+\left[f(x_{i+1})+f(x_i)\right] \\\end{split}\]</div>
+<p>Special Case (Equaliy spaced base points)</p>
+<div class="math notranslate nohighlight">
+\[x_{i+1}-x_i = h \quad \forall i\]</div>
+<div class="math notranslate nohighlight">
+\[I \approx  h \left\{ \frac{1}{2} \left[f(x_0)+f(x_n)\right] +
+  \sum\limits_{i=1}^{n-1} f(x_i) \right\}\]</div>
+<p class="rubric">Example</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}I &amp;= \int\limits_a^b f(x) \,\mathrm{d}x \\
+f(x) &amp;= x^2 \\
+a &amp;= 0 \\
+b &amp;= 1\end{split}\]</div>
+<p>analytical solution</p>
+<div class="math notranslate nohighlight">
+\[I = \int\limits_{0}^{1} x^2 \,\mathrm{d}x
+  = \left. \frac{1}{3} x^3 \right\vert_0^1
+  = \frac{1}{3}\]</div>
+<p>numerical solution</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span><span class="p">(</span><span class="n">x</span><span class="p">):</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">trapez</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+<span class="go">0.5</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">trapez</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
+<span class="go">0.3350000000000001</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">trapez</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
+<span class="go">0.33335000000000004</span>
+</pre></div>
+</div>
+</dd></dl>
+
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