add function integration, tests and docs
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<title>numerical package — pylib 2019.5.19 documentation</title>
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<div class="section" id="numerical-package">
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<h1>numerical package<a class="headerlink" href="#numerical-package" title="Permalink to this headline">¶</a></h1>
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<div class="section" id="submodules">
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<h2>Submodules<a class="headerlink" href="#submodules" title="Permalink to this headline">¶</a></h2>
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</div>
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<div class="section" id="module-numerical.fit">
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<span id="numerical-fit-module"></span><h2>numerical.fit module<a class="headerlink" href="#module-numerical.fit" title="Permalink to this headline">¶</a></h2>
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<p>Function and approximation.</p>
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<span class="target" id="module-fit"></span><dl class="function">
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<dt id="numerical.fit.gauss">
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<code class="descname">gauss</code><span class="sig-paren">(</span><em>x</em>, <em>*p</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/numerical/fit.html#gauss"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numerical.fit.gauss" title="Permalink to this definition">¶</a></dt>
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<dd><p>Gauss distribution function.</p>
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<div class="math notranslate nohighlight">
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\[f(x)=ae^{-(x-b)^{2}/(2c^{2})}\]</div>
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<dl class="field-list simple">
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<dt class="field-odd">Parameters</dt>
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<dd class="field-odd"><ul class="simple">
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<li><p><strong>x</strong> (<em>int</em><em> or </em><em>float</em><em> or </em><em>list</em><em> or </em><em>numpy.ndarray</em>) – positions where the gauss function will be calculated</p></li>
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<li><p><strong>p</strong> (<em>list</em>) – <p>gauss parameters [a, b, c, d]:</p>
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<ul>
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<li><p>a – amplitude (<span class="math notranslate nohighlight">\(\int y \,\mathrm{d}x=1 \Leftrightarrow a=1/(c\sqrt{2\pi})\)</span> )</p></li>
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<li><p>b – expected value <span class="math notranslate nohighlight">\(\mu\)</span> (position of maximum, default = 0)</p></li>
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<li><p>c – standard deviation <span class="math notranslate nohighlight">\(\sigma\)</span> (variance <span class="math notranslate nohighlight">\(\sigma^2=c^2\)</span>)</p></li>
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<li><p>d – vertical offset (default = 0)</p></li>
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</ul>
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</p></li>
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</ul>
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</dd>
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<dt class="field-even">Returns</dt>
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<dd class="field-even"><p>gauss values at given positions x</p>
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</dd>
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<dt class="field-odd">Return type</dt>
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<dd class="field-odd"><p>numpy.ndarray</p>
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</dd>
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</dl>
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</dd></dl>
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<dl class="function">
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<dt id="numerical.fit.gauss_fit">
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<code class="descname">gauss_fit</code><span class="sig-paren">(</span><em>x</em>, <em>y</em>, <em>e=None</em>, <em>x_fit=None</em>, <em>verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/numerical/fit.html#gauss_fit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numerical.fit.gauss_fit" title="Permalink to this definition">¶</a></dt>
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<dd><p>Fit Gauss distribution function to data.</p>
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<dl class="field-list simple">
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<dt class="field-odd">Parameters</dt>
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<dd class="field-odd"><ul class="simple">
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<li><p><strong>x</strong> (<em>int</em><em> or </em><em>float</em><em> or </em><em>list</em><em> or </em><em>numpy.ndarray</em>) – positions</p></li>
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<li><p><strong>y</strong> (<em>int</em><em> or </em><em>float</em><em> or </em><em>list</em><em> or </em><em>numpy.ndarray</em>) – values</p></li>
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<li><p><strong>e</strong> (<em>int</em><em> or </em><em>float</em><em> or </em><em>list</em><em> or </em><em>numpy.ndarray</em>) – error values (default = None)</p></li>
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<li><p><strong>x_fit</strong> (<em>int</em><em> or </em><em>float</em><em> or </em><em>list</em><em> or </em><em>numpy.ndarray</em>) – positions of fitted function (default = None, if None then x
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is used)</p></li>
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<li><p><strong>verbose</strong> (<em>bool</em>) – verbose information (default = False)</p></li>
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</ul>
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</dd>
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<dt class="field-even">Returns</dt>
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<dd class="field-even"><p><ul class="simple">
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<li><p>numpy.ndarray – fitted values (y_fit)</p></li>
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<li><p>numpy.ndarray – parameters of gauss distribution function (popt:
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amplitude a, expected value <span class="math notranslate nohighlight">\(\mu\)</span>, standard deviation
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<span class="math notranslate nohighlight">\(\sigma\)</span>, vertical offset d)</p></li>
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<li><p>numpy.float64 – full width at half maximum (FWHM)</p></li>
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</ul>
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</p>
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</dd>
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<dt class="field-odd">Return type</dt>
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<dd class="field-odd"><p>tuple</p>
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</dd>
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</dl>
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<div class="admonition seealso">
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<p class="admonition-title">See also</p>
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<p><a class="reference internal" href="#numerical.fit.gauss" title="numerical.fit.gauss"><code class="xref py py-meth docutils literal notranslate"><span class="pre">gauss()</span></code></a></p>
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</div>
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</dd></dl>
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</div>
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<div class="section" id="module-numerical.integration">
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<span id="numerical-integration-module"></span><h2>numerical.integration module<a class="headerlink" href="#module-numerical.integration" title="Permalink to this headline">¶</a></h2>
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<p>Numerical integration, numerical quadrature.</p>
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<p>de: numerische Integration, numerische Quadratur.</p>
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<dl class="field-list simple">
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<dt class="field-odd">Date</dt>
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<dd class="field-odd"><p>2015-10-15</p>
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</dd>
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</dl>
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<span class="target" id="module-integration"></span><dl class="function">
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<dt id="numerical.integration.trapez">
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<code class="descname">trapez</code><span class="sig-paren">(</span><em>f</em>, <em>a=0</em>, <em>b=1</em>, <em>N=10</em>, <em>x=None</em>, <em>verbose=False</em>, <em>save_values=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/numerical/integration.html#trapez"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numerical.integration.trapez" title="Permalink to this definition">¶</a></dt>
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<dd><p>Integration of <span class="math notranslate nohighlight">\(f(x)\)</span> using the trapezoidal rule
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(Simpson’s rule, Kepler’s rule).</p>
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<p>de: Trapezregel, Simpsonregel (Thomas Simpson), Keplersche
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Fassregel (Johannes Kepler)</p>
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<dl class="field-list simple">
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<dt class="field-odd">Parameters</dt>
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<dd class="field-odd"><ul class="simple">
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<li><p><strong>f</strong> (<em>function</em><em> or </em><em>list</em>) – function to integrate.</p></li>
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<li><p><strong>a</strong> (<em>float</em>) – lower limit of integration (default = 0).</p></li>
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<li><p><strong>b</strong> (<em>float</em>) – upper limit of integration (default = 1).</p></li>
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<li><p><strong>N</strong> (<em>int</em>) – specify the number of subintervals.</p></li>
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<li><p><strong>x</strong> (<em>list</em>) – variable of integration, necessary if f is a list
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(default = None).</p></li>
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<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
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</ul>
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</dd>
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<dt class="field-even">Returns</dt>
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<dd class="field-even"><p>the definite integral as approximated by trapezoidal
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rule.</p>
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</dd>
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<dt class="field-odd">Return type</dt>
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<dd class="field-odd"><p>float</p>
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</dd>
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</dl>
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<p>The trapezoidal rule approximates the integral by the area of a
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trapezoid with base h=b-a and sides equal to the values of the
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integrand at the two end points.</p>
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<div class="math notranslate nohighlight">
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\[f_n(x) = f(a)+\frac{f(b)-f(a)}{b-a}(x-a)\]</div>
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<div class="math notranslate nohighlight">
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\[\begin{split}I &= \int\limits_a^b f(x) \,\mathrm{d}x \\
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I &\approx \int\limits_a^b f_n(x) \,\mathrm{d}x \\
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&= \int\limits_a^b
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\left( f(a)+\frac{f(b)-f(a)}{b-a}(x-a) \right)
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\mathrm{d}x \\
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&= \left.\left( f(a)-a\frac{f(b)-f(a)}{b-a} \right)
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x \right\vert_a^b +
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\left. \frac{f(b)-f(a)}{b-a} \frac{x^2}{2}
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\right\vert_a^b \\
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&= \frac{b-a}{2}\left[f(a)+f(b)\right]\end{split}\]</div>
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<p>The composite trapezium rule. If the interval is divided into n
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segments (not necessarily equal)</p>
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<div class="math notranslate nohighlight">
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\[a = x_0 \leq x_1 \leq x_2 \leq \ldots \leq x_n = b\]</div>
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<div class="math notranslate nohighlight">
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\[\begin{split}I &\approx \sum\limits_{i=0}^{n-1} \frac{1}{2} (x_{i+1}-x_i)
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\left[f(x_{i+1})+f(x_i)\right] \\\end{split}\]</div>
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<p>Special Case (Equaliy spaced base points)</p>
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<div class="math notranslate nohighlight">
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\[x_{i+1}-x_i = h \quad \forall i\]</div>
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<div class="math notranslate nohighlight">
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\[I \approx h \left\{ \frac{1}{2} \left[f(x_0)+f(x_n)\right] +
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\sum\limits_{i=1}^{n-1} f(x_i) \right\}\]</div>
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<p class="rubric">Example</p>
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<div class="math notranslate nohighlight">
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\[\begin{split}I &= \int\limits_a^b f(x) \,\mathrm{d}x \\
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f(x) &= x^2 \\
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a &= 0 \\
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b &= 1\end{split}\]</div>
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<p>analytical solution</p>
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<div class="math notranslate nohighlight">
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\[I = \int\limits_{0}^{1} x^2 \,\mathrm{d}x
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= \left. \frac{1}{3} x^3 \right\vert_0^1
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= \frac{1}{3}\]</div>
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<p>numerical solution</p>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </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>
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<span class="gp">>>> </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>
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<span class="go">0.5</span>
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<span class="gp">>>> </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>
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<span class="go">0.3350000000000001</span>
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<span class="gp">>>> </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>
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<span class="go">0.33335000000000004</span>
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</pre></div>
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</div>
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</dd></dl>
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</div>
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<div class="section" id="module-numerical.ode">
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<span id="numerical-ode-module"></span><h2>numerical.ode module<a class="headerlink" href="#module-numerical.ode" title="Permalink to this headline">¶</a></h2>
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<p>Numerical solver of ordinary differential equations.</p>
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<p>Solves the initial value problem for systems of first order ordinary differential
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equations.</p>
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<dl class="field-list simple">
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<dt class="field-odd">Date</dt>
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<dd class="field-odd"><p>2015-09-21</p>
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</dd>
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</dl>
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<span class="target" id="module-ode"></span><dl class="function">
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<dt id="numerical.ode.dxdt_Dt">
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<code class="descname">dxdt_Dt</code><span class="sig-paren">(</span><em>f</em>, <em>x</em>, <em>t</em>, <em>Dt</em>, <em>*p</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/numerical/ode.html#dxdt_Dt"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numerical.ode.dxdt_Dt" title="Permalink to this definition">¶</a></dt>
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<dd><dl class="field-list simple">
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<dt class="field-odd">Parameters</dt>
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<dd class="field-odd"><ul class="simple">
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<li><p><strong>f</strong> (<em>function</em>) – <span class="math notranslate nohighlight">\(f = \dot{x}\)</span></p></li>
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<li><p><strong>Dt</strong> – <span class="math notranslate nohighlight">\(\Delta{t}\)</span></p></li>
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</ul>
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</dd>
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<dt class="field-even">Returns</dt>
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<dd class="field-even"><p><span class="math notranslate nohighlight">\(\Delta x = \dot{x} \Delta t\)</span></p>
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</dd>
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</dl>
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</dd></dl>
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<dl class="function">
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<dt id="numerical.ode.e1">
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<code class="descname">e1</code><span class="sig-paren">(</span><em>f</em>, <em>x0</em>, <em>t</em>, <em>*p</em>, <em>verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/numerical/ode.html#e1"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numerical.ode.e1" title="Permalink to this definition">¶</a></dt>
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<dd><p>Explicit first-order method /
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(standard, or forward) Euler method /
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Runge-Kutta 1st order method.</p>
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<p>de:
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||||
Euler’sche Polygonzugverfahren / explizite Euler-Verfahren /
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Euler-Cauchy-Verfahren / Euler-vorwärts-Verfahren</p>
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<dl class="field-list simple">
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<dt class="field-odd">Parameters</dt>
|
||||
<dd class="field-odd"><ul class="simple">
|
||||
<li><p><strong>f</strong> (<em>function</em>) – the function to solve</p></li>
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||||
<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
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||||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
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||||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter, …)</p></li>
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||||
<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
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</ul>
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</dd>
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</dl>
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<p>Approximate the solution of the initial value problem</p>
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<div class="math notranslate nohighlight">
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\[\begin{split}\dot{x} &= f(t,x) \\
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||||
x(t_0) &= x_0\end{split}\]</div>
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<p>Choose a value h for the size of every step and set</p>
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||||
<div class="math notranslate nohighlight">
|
||||
\[t_i = t_0 + i h ~,\quad i=1,2,\ldots,n\]</div>
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||||
<p>The derivative of the solution is approximated as the forward difference
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||||
equation</p>
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<div class="math notranslate nohighlight">
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||||
\[\dot{x}_i = f(t_i, x_i) = \frac{x_{i+1} - x_i}{t_{i+1}-t_i}\]</div>
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<p>Therefore one step <span class="math notranslate nohighlight">\(h\)</span> of the Euler method from <span class="math notranslate nohighlight">\(t_i\)</span> to
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<span class="math notranslate nohighlight">\(t_{i+1}\)</span> is</p>
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||||
<div class="math notranslate nohighlight">
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||||
\[\begin{split}x_{i+1} &= x_i + (t_{i+1}-t_i) f(t_i, x_i) \\
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x_{i+1} &= x_i + h f(t_i, x_i) \\\end{split}\]</div>
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<p>Example 1:</p>
|
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<div class="math notranslate nohighlight">
|
||||
\[\begin{split}m\ddot{u} + d\dot{u} + ku = f(t) \\
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\ddot{u} = m^{-1}(f(t) - d\dot{u} - ku) \\\end{split}\]</div>
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<p>with</p>
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<div class="math notranslate nohighlight">
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||||
\[\begin{split}x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\
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x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\\end{split}\]</div>
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||||
<p>becomes</p>
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||||
<div class="math notranslate nohighlight">
|
||||
\[\begin{split}\dot{x}_1 &= x_2 \\
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||||
\dot{x}_2 &= m^{-1}(f(t) - d x_2 - k x_1) \\\end{split}\]</div>
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||||
<p>or</p>
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||||
<div class="math notranslate nohighlight">
|
||||
\[\begin{split}\dot{x} &= f(t,x) \\
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||||
\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &=
|
||||
\begin{bmatrix} x_2 \\ m^{-1}(f(t) - d x_2 - k x_1) \end{bmatrix} \\
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||||
&=
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||||
\begin{bmatrix} 0 \\ m^{-1} f(t) \end{bmatrix} +
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||||
\begin{bmatrix} 0 & 1 \\ -m^{-1} k & -m^{-1} d \end{bmatrix}
|
||||
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\end{split}\]</div>
|
||||
<p>Example 2:</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[\begin{split}m(u)\ddot{u} + d(u,\dot{u})\dot{u} + k(u)u = f(t) \\
|
||||
\ddot{u} = m^{-1}(u)(f(t) - d(u,\dot{u})\dot{u} - k(u)u) \\\end{split}\]</div>
|
||||
<p>with</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[\begin{split}x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\
|
||||
x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\\end{split}\]</div>
|
||||
<p>becomes</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[\begin{split}\dot{x}_1 &= x_2 \\
|
||||
\dot{x}_2 &= m^{-1}(x_1)(f(t) - d(x_1,x_2) x_2 - k(x_1) x_1) \\\end{split}\]</div>
|
||||
<p>or</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[\begin{split}\dot{x} &= f(t,x) \\
|
||||
\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &=
|
||||
\begin{bmatrix} x_2 \\ m^{-1}(x_1)(f(t) - d(x_1,x_2) x_2 - k(x_1) x_1) \end{bmatrix} \\
|
||||
&=
|
||||
\begin{bmatrix} 0 \\ m^{-1}(x_1) f(t) \end{bmatrix} +
|
||||
\begin{bmatrix} 0 & 1 \\ -m^{-1}(x_1) k(x_1) & -m^{-1} d(x_1,x_2) \end{bmatrix}
|
||||
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\end{split}\]</div>
|
||||
<p>The Euler method is a first-order method,
|
||||
which means that the local error (error per step) is proportional to the
|
||||
square of the step size, and the global error (error at a given time) is
|
||||
proportional to the step size.</p>
|
||||
</dd></dl>
|
||||
|
||||
<dl class="function">
|
||||
<dt id="numerical.ode.e2">
|
||||
<code class="descname">e2</code><span class="sig-paren">(</span><em>f</em>, <em>x0</em>, <em>t</em>, <em>*p</em>, <em>verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/numerical/ode.html#e2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numerical.ode.e2" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Explicit second-order method / Runge-Kutta 2nd order method.</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>) – the function to solve</p></li>
|
||||
<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
|
||||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter, …)</p></li>
|
||||
<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
|
||||
</ul>
|
||||
</dd>
|
||||
</dl>
|
||||
</dd></dl>
|
||||
|
||||
<dl class="function">
|
||||
<dt id="numerical.ode.e4">
|
||||
<code class="descname">e4</code><span class="sig-paren">(</span><em>f</em>, <em>x0</em>, <em>t</em>, <em>*p</em>, <em>verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/numerical/ode.html#e4"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numerical.ode.e4" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Explicit fourth-order method / Runge-Kutta 4th order method.</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>) – the function to solve</p></li>
|
||||
<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
|
||||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter, …)</p></li>
|
||||
<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
|
||||
</ul>
|
||||
</dd>
|
||||
</dl>
|
||||
</dd></dl>
|
||||
|
||||
<dl class="function">
|
||||
<dt id="numerical.ode.fixed_point_iteration">
|
||||
<code class="descname">fixed_point_iteration</code><span class="sig-paren">(</span><em>f</em>, <em>xi</em>, <em>t</em>, <em>max_iterations=1000</em>, <em>tol=1e-09</em>, <em>verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/numerical/ode.html#fixed_point_iteration"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numerical.ode.fixed_point_iteration" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><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>) – the function to iterate <span class="math notranslate nohighlight">\(f = \Delta{x}(t)\)</span></p></li>
|
||||
<li><p><strong>xi</strong> (<em>list</em>) – initial condition <span class="math notranslate nohighlight">\(x_i\)</span></p></li>
|
||||
<li><p><strong>t</strong> (<em>float</em>) – time <span class="math notranslate nohighlight">\(t\)</span></p></li>
|
||||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter, …)</p></li>
|
||||
<li><p><strong>max_iterations</strong> (<em>int</em>) – maximum number of iterations</p></li>
|
||||
<li><p><strong>tol</strong> (<em>float</em>) – tolerance against residuum (default = 1e-9)</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><span class="math notranslate nohighlight">\(x_{i+1}\)</span></p>
|
||||
</dd>
|
||||
</dl>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[x_{i+1} = x_i + \Delta x\]</div>
|
||||
<div class="admonition seealso">
|
||||
<p class="admonition-title">See also</p>
|
||||
<p><a class="reference internal" href="#numerical.ode.dxdt_Dt" title="numerical.ode.dxdt_Dt"><code class="xref py py-meth docutils literal notranslate"><span class="pre">dxdt_Dt()</span></code></a> for <span class="math notranslate nohighlight">\(\Delta x\)</span></p>
|
||||
</div>
|
||||
</dd></dl>
|
||||
|
||||
<dl class="function">
|
||||
<dt id="numerical.ode.i1">
|
||||
<code class="descname">i1</code><span class="sig-paren">(</span><em>f</em>, <em>x0</em>, <em>t</em>, <em>*p</em>, <em>max_iterations=1000</em>, <em>tol=1e-09</em>, <em>verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/numerical/ode.html#i1"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numerical.ode.i1" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Implicite first-order method / backward Euler method.</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>) – the function to solve</p></li>
|
||||
<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
|
||||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter, …)</p></li>
|
||||
<li><p><strong>max_iterations</strong> (<em>int</em>) – maximum number of iterations</p></li>
|
||||
<li><p><strong>tol</strong> (<em>float</em>) – tolerance against residuum (default = 1e-9)</p></li>
|
||||
<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
|
||||
</ul>
|
||||
</dd>
|
||||
</dl>
|
||||
<p>The backward Euler method has order one and is A-stable.</p>
|
||||
</dd></dl>
|
||||
|
||||
<dl class="function">
|
||||
<dt id="numerical.ode.i1n">
|
||||
<code class="descname">i1n</code><span class="sig-paren">(</span><em>f</em>, <em>x0</em>, <em>t</em>, <em>*p</em>, <em>max_iterations=1000</em>, <em>tol=1e-09</em>, <em>verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/numerical/ode.html#i1n"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numerical.ode.i1n" title="Permalink to this definition">¶</a></dt>
|
||||
<dd></dd></dl>
|
||||
|
||||
<dl class="function">
|
||||
<dt id="numerical.ode.newmark_newtonraphson">
|
||||
<code class="descname">newmark_newtonraphson</code><span class="sig-paren">(</span><em>f</em>, <em>x0</em>, <em>xp0</em>, <em>xpp0</em>, <em>t</em>, <em>*p</em>, <em>gamma=0.5</em>, <em>beta=0.25</em>, <em>max_iterations=1000</em>, <em>tol=1e-09</em>, <em>verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/numerical/ode.html#newmark_newtonraphson"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numerical.ode.newmark_newtonraphson" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Newmark method.</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>) – the function to solve</p></li>
|
||||
<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
|
||||
<li><p><strong>xp0</strong> (<em>list</em>) – initial condition</p></li>
|
||||
<li><p><strong>xpp0</strong> (<em>list</em>) – initial condition</p></li>
|
||||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter, …)</p></li>
|
||||
<li><p><strong>gamma</strong> (<em>float</em>) – newmark parameter for velocity (default = 0.5)</p></li>
|
||||
<li><p><strong>beta</strong> (<em>float</em>) – newmark parameter for displacement (default = 0.25)</p></li>
|
||||
<li><p><strong>max_iterations</strong> (<em>int</em>) – maximum number of iterations</p></li>
|
||||
<li><p><strong>tol</strong> (<em>float</em>) – tolerance against residuum (default = 1e-9)</p></li>
|
||||
<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
|
||||
</ul>
|
||||
</dd>
|
||||
</dl>
|
||||
</dd></dl>
|
||||
|
||||
<dl class="function">
|
||||
<dt id="numerical.ode.newmark_newtonraphson_rdk">
|
||||
<code class="descname">newmark_newtonraphson_rdk</code><span class="sig-paren">(</span><em>fnm</em>, <em>x0</em>, <em>xp0</em>, <em>xpp0</em>, <em>t</em>, <em>*p</em>, <em>gamma=0.5</em>, <em>beta=0.25</em>, <em>maxIterations=1000</em>, <em>tol=1e-09</em>, <em>verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/numerical/ode.html#newmark_newtonraphson_rdk"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numerical.ode.newmark_newtonraphson_rdk" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Newmark method.</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>) – the function to solve</p></li>
|
||||
<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
|
||||
<li><p><strong>xp0</strong> (<em>list</em>) – initial condition</p></li>
|
||||
<li><p><strong>xpp0</strong> (<em>list</em>) – initial condition</p></li>
|
||||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter, …)</p></li>
|
||||
<li><p><strong>gamma</strong> (<em>float</em>) – newmark parameter for velocity (default = 0.5)</p></li>
|
||||
<li><p><strong>beta</strong> (<em>float</em>) – newmark parameter for displacement (default = 0.25)</p></li>
|
||||
<li><p><strong>max_iterations</strong> (<em>int</em>) – maximum number of iterations</p></li>
|
||||
<li><p><strong>tol</strong> (<em>float</em>) – tolerance against residuum (default = 1e-9)</p></li>
|
||||
<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
|
||||
</ul>
|
||||
</dd>
|
||||
</dl>
|
||||
</dd></dl>
|
||||
|
||||
</div>
|
||||
<div class="section" id="module-numerical.ode_model">
|
||||
<span id="numerical-ode-model-module"></span><h2>numerical.ode_model module<a class="headerlink" href="#module-numerical.ode_model" title="Permalink to this headline">¶</a></h2>
|
||||
<p>Mathmatical models governed by ordinary differential equations.</p>
|
||||
<p>Describes initial value problems as systems of first order ordinary differential
|
||||
equations.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Date</dt>
|
||||
<dd class="field-odd"><p>2019-05-25</p>
|
||||
</dd>
|
||||
</dl>
|
||||
<span class="target" id="module-ode_model"></span><dl class="function">
|
||||
<dt id="numerical.ode_model.disk">
|
||||
<code class="descname">disk</code><span class="sig-paren">(</span><em>x</em>, <em>t</em>, <em>*p</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/numerical/ode_model.html#disk"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numerical.ode_model.disk" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Rotation of an eccentric disk.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
<dd class="field-odd"><ul class="simple">
|
||||
<li><p><strong>x</strong> (<em>list</em>) – values of the function</p></li>
|
||||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||||
<li><p><strong>*p</strong> – <p>parameters of the function</p>
|
||||
<ul>
|
||||
<li><p>diameter</p></li>
|
||||
<li><p>eccentricity</p></li>
|
||||
<li><p>torque</p></li>
|
||||
</ul>
|
||||
</p></li>
|
||||
</ul>
|
||||
</dd>
|
||||
</dl>
|
||||
</dd></dl>
|
||||
|
||||
<dl class="function">
|
||||
<dt id="numerical.ode_model.disk_nm">
|
||||
<code class="descname">disk_nm</code><span class="sig-paren">(</span><em>xn</em>, <em>xpn</em>, <em>xppn</em>, <em>t</em>, <em>*p</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/numerical/ode_model.html#disk_nm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numerical.ode_model.disk_nm" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Rotation of an eccentric disk.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
<dd class="field-odd"><ul class="simple">
|
||||
<li><p><strong>xn</strong> (<em>list</em>) – values of the function</p></li>
|
||||
<li><p><strong>xpn</strong> (<em>list</em>) – first derivative values of the function</p></li>
|
||||
<li><p><strong>xppn</strong> (<em>list</em>) – second derivative values of the function</p></li>
|
||||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||||
<li><p><strong>*p</strong> – <p>parameters of the function</p>
|
||||
<ul>
|
||||
<li><p>diameter</p></li>
|
||||
<li><p>eccentricity</p></li>
|
||||
<li><p>torque</p></li>
|
||||
</ul>
|
||||
</p></li>
|
||||
</ul>
|
||||
</dd>
|
||||
</dl>
|
||||
</dd></dl>
|
||||
|
||||
<dl class="function">
|
||||
<dt id="numerical.ode_model.disk_nmmdk">
|
||||
<code class="descname">disk_nmmdk</code><span class="sig-paren">(</span><em>xn</em>, <em>xpn</em>, <em>xppn</em>, <em>t</em>, <em>*p</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/numerical/ode_model.html#disk_nmmdk"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numerical.ode_model.disk_nmmdk" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Rotation of an eccentric disk.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
<dd class="field-odd"><ul class="simple">
|
||||
<li><p><strong>xn</strong> (<em>list</em>) – values of the function</p></li>
|
||||
<li><p><strong>xpn</strong> (<em>list</em>) – derivative values of the function</p></li>
|
||||
<li><p><strong>xppn</strong> (<em>list</em>) – second derivative values of the function</p></li>
|
||||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||||
<li><p><strong>*p</strong> – <p>parameters of the function</p>
|
||||
<ul>
|
||||
<li><p>diameter</p></li>
|
||||
<li><p>eccentricity</p></li>
|
||||
<li><p>torque</p></li>
|
||||
</ul>
|
||||
</p></li>
|
||||
</ul>
|
||||
</dd>
|
||||
</dl>
|
||||
</dd></dl>
|
||||
|
||||
</div>
|
||||
<div class="section" id="module-numerical">
|
||||
<span id="module-contents"></span><h2>Module contents<a class="headerlink" href="#module-numerical" title="Permalink to this headline">¶</a></h2>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
||||
</div>
|
||||
|
||||
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|
||||
</div>
|
||||
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|
||||
<div class="sphinxsidebarwrapper">
|
||||
<h1 class="logo"><a href="index.html">pylib</a></h1>
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
<h3>Navigation</h3>
|
||||
|
||||
<div class="relations">
|
||||
<h3>Related Topics</h3>
|
||||
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|
||||
<li><a href="index.html">Documentation overview</a><ul>
|
||||
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||||
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||||
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||||
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||||
©2019, Daniel Weschke.
|
||||
|
||||
|
|
||||
Powered by <a href="http://sphinx-doc.org/">Sphinx 2.0.1</a>
|
||||
& <a href="https://github.com/bitprophet/alabaster">Alabaster 0.7.12</a>
|
||||
|
||||
|
|
||||
<a href="_sources/numerical.rst.txt"
|
||||
rel="nofollow">Page source</a>
|
||||
</div>
|
||||
|
||||
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
Reference in New Issue
Block a user