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add mathematical functions module

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update docs and clean up code
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Daniel Weschke
2019-10-27 15:02:44 +01:00
parent b474188d1a
commit b566c643db
21 changed files with 592 additions and 71 deletions
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docs/build/html/_modules/numerical/integration.html vendored
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@@ -51,7 +51,7 @@
<span class="kn">from</span> <span class="nn">numpy</span> <span class="k">import</span> <span class="n">linspace</span><span class="p">,</span> <span class="n">trapz</span><span class="p">,</span> <span class="n">zeros</span>
<div class="viewcode-block" id="trapez"><a class="viewcode-back" href="../../numerical.html#numerical.integration.trapez">[docs]</a><span class="k">def</span> <span class="nf">trapez</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
<span class="n">save_values</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
<span class="n">save_values</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
<span class="sa">r</span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> Integration of :math:`f(x)` using the trapezoidal rule</span>
<span class="sd"> (Simpson&#39;s rule, Kepler&#39;s rule).</span>
@@ -122,14 +122,14 @@
<span class="sd"> f(x) &amp;= x^2 \\</span>
<span class="sd"> a &amp;= 0 \\</span>
<span class="sd"> b &amp;= 1</span>
<span class="sd"> </span>
<span class="sd"> analytical solution</span>
<span class="sd"> .. math::</span>
<span class="sd"> I = \int\limits_{0}^{1} x^2 \,\mathrm{d}x</span>
<span class="sd"> = \left. \frac{1}{3} x^3 \right\vert_0^1</span>
<span class="sd"> = \frac{1}{3}</span>
<span class="sd"> </span>
<span class="sd"> numerical solution</span>
<span class="sd"> &gt;&gt;&gt; f = lambda(x): x**2</span>
@@ -142,30 +142,30 @@
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
<span class="c1"># f is function or list</span>
<span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s1">&#39;__call__&#39;</span><span class="p">):</span>
<span class="c1"># h width of each subinterval</span>
<span class="n">h</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">N</span>
<span class="k">if</span> <span class="n">callable</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
<span class="c1"># h width of each subinterval</span>
<span class="n">h</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">N</span>
<span class="c1"># x variable of integration</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">linspace</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">N</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="k">if</span> <span class="n">save_values</span><span class="p">:</span>
<span class="c1"># ff contribution from the points</span>
<span class="n">ff</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
<span class="n">ff</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
<span class="n">T</span> <span class="o">=</span> <span class="p">(</span><span class="n">ff</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="mf">2.</span><span class="o">+</span><span class="nb">sum</span><span class="p">(</span><span class="n">ff</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="n">N</span><span class="p">])</span><span class="o">+</span><span class="n">ff</span><span class="p">[</span><span class="n">N</span><span class="p">]</span><span class="o">/</span><span class="mf">2.</span><span class="p">)</span><span class="o">*</span><span class="n">h</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">TL</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">TR</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">N</span><span class="p">])</span>
<span class="n">TI</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="p">):</span>
<span class="n">TI</span> <span class="o">=</span> <span class="n">TI</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
<span class="n">T</span> <span class="o">=</span> <span class="p">(</span><span class="n">TL</span><span class="o">/</span><span class="mf">2.</span><span class="o">+</span><span class="n">TI</span><span class="o">+</span><span class="n">TR</span><span class="o">/</span><span class="mf">2.</span><span class="p">)</span><span class="o">*</span><span class="n">h</span>
<span class="c1"># x variable of integration</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">linspace</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">N</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="k">if</span> <span class="n">save_values</span><span class="p">:</span>
<span class="c1"># ff contribution from the points</span>
<span class="n">ff</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
<span class="n">ff</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
<span class="n">T</span> <span class="o">=</span> <span class="p">(</span><span class="n">ff</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="mf">2.</span><span class="o">+</span><span class="nb">sum</span><span class="p">(</span><span class="n">ff</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="n">N</span><span class="p">])</span><span class="o">+</span><span class="n">ff</span><span class="p">[</span><span class="n">N</span><span class="p">]</span><span class="o">/</span><span class="mf">2.</span><span class="p">)</span><span class="o">*</span><span class="n">h</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">TL</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">TR</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">N</span><span class="p">])</span>
<span class="n">TI</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="p">):</span>
<span class="n">TI</span> <span class="o">=</span> <span class="n">TI</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
<span class="n">T</span> <span class="o">=</span> <span class="p">(</span><span class="n">TL</span><span class="o">/</span><span class="mf">2.</span><span class="o">+</span><span class="n">TI</span><span class="o">+</span><span class="n">TR</span><span class="o">/</span><span class="mf">2.</span><span class="p">)</span><span class="o">*</span><span class="n">h</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span>
<span class="n">T</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">T</span> <span class="o">=</span> <span class="n">T</span> <span class="o">+</span> <span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="p">])</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">f</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span>
<span class="n">T</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">T</span> <span class="o">=</span> <span class="n">T</span> <span class="o">+</span> <span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="p">])</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">f</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
<span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="n">T</span><span class="p">)</span>
@@ -178,12 +178,12 @@
<span class="n">trapez</span><span class="p">(</span><span class="n">func</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="mf">1e6</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="c1">#print(trapz(func, linspace(0,1,10)))</span>
<span class="n">trapez</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">4</span><span class="p">,</span><span class="mi">9</span><span class="p">],</span> <span class="n">x</span><span class="o">=</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">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">trapez</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">4</span><span class="p">,</span> <span class="mi">9</span><span class="p">],</span> <span class="n">x</span><span class="o">=</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">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="c1">#print(trapz([0,1,4,9]))</span>
<span class="n">trapez</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="n">x</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">trapez</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="n">x</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">trapez</span><span class="p">([</span><span class="o">-</span><span class="mi">1</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="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">x</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</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="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">trapez</span><span class="p">([</span><span class="o">-</span><span class="mi">1</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="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">x</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</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="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
</pre></div>
</div>