add function integration, tests and docs
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*.egg-info/
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# Sphinx documentation
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docs/build/.doctrees/
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docs/build/doctrees/
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# Sphinx build info version 1
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# This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done.
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config: 94375f4299332632f508e2242b7c30d8
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config: aaf67f6f94ce2e6ce3750a4b226f6461
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tags: 645f666f9bcd5a90fca523b33c5a78b7
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179
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<!DOCTYPE html>
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<html xmlns="http://www.w3.org/1999/xhtml">
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<head>
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<meta charset="utf-8" />
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<title>data — pylib 2019.5.19 documentation</title>
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<link rel="stylesheet" href="../_static/alabaster.css" type="text/css" />
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<link rel="stylesheet" href="../_static/pygments.css" type="text/css" />
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<link rel="stylesheet" href="../_static/custom.css" type="text/css" />
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<script type="text/javascript" id="documentation_options" data-url_root="../" src="../_static/documentation_options.js"></script>
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<script type="text/javascript" src="../_static/jquery.js"></script>
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<script type="text/javascript" src="../_static/underscore.js"></script>
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<script type="text/javascript" src="../_static/doctools.js"></script>
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<script type="text/javascript" src="../_static/language_data.js"></script>
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<script async="async" type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-AMS-MML_HTMLorMML"></script>
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<link rel="index" title="Index" href="../genindex.html" />
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<link rel="search" title="Search" href="../search.html" />
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<link rel="stylesheet" href="../_static/custom.css" type="text/css" />
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<meta name="viewport" content="width=device-width, initial-scale=0.9, maximum-scale=0.9" />
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</head><body>
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<div class="document">
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<div class="documentwrapper">
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<div class="bodywrapper">
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<div class="body" role="main">
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<h1>Source code for data</h1><div class="highlight"><pre>
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<span></span><span class="ch">#!/usr/bin/env python</span>
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<span class="c1"># -*- coding: utf-8 -*-</span>
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<span class="sd">"""Read and write data to or from file.</span>
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<span class="sd">.. module:: data</span>
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<span class="sd"> :platform: *nix, Windows</span>
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<span class="sd"> :synopsis: Handle data files.</span>
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<span class="sd">.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de></span>
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<span class="sd">"""</span>
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<span class="kn">from</span> <span class="nn">__future__</span> <span class="k">import</span> <span class="n">print_function</span>
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<span class="kn">import</span> <span class="nn">pickle</span>
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<div class="viewcode-block" id="data_read"><a class="viewcode-back" href="../data.html#data.data_read">[docs]</a><span class="k">def</span> <span class="nf">data_read</span><span class="p">(</span><span class="n">file_name</span><span class="p">,</span> <span class="n">x_column</span><span class="p">,</span> <span class="n">y_column</span><span class="p">):</span>
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<span class="sd">"""Read ascii data file.</span>
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<span class="sd"> :param filename: file to read</span>
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<span class="sd"> :type filename: str</span>
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<span class="sd"> :param x_column: column index for the x data (first column is 0)</span>
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<span class="sd"> :type x_column: int</span>
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<span class="sd"> :param y_column: column index for the y data (first column is 0)</span>
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<span class="sd"> :type y_column: int</span>
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<span class="sd"> :returns: x and y</span>
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<span class="sd"> :rtype: tuple(list, list)</span>
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<span class="sd"> """</span>
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<span class="kn">import</span> <span class="nn">re</span>
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<span class="n">file</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">file_name</span><span class="p">)</span>
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<span class="n">x</span> <span class="o">=</span> <span class="p">[]</span>
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<span class="n">y</span> <span class="o">=</span> <span class="p">[]</span>
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<span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">file</span><span class="p">:</span>
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<span class="n">fields</span> <span class="o">=</span> <span class="n">re</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="sa">r</span><span class="s1">'\s+'</span><span class="p">,</span> <span class="n">row</span><span class="o">.</span><span class="n">strip</span><span class="p">())</span>
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<span class="c1">#print(filds)</span>
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<span class="n">x</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">fields</span><span class="p">[</span><span class="n">x_column</span><span class="p">]))</span>
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<span class="n">y</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">fields</span><span class="p">[</span><span class="n">y_column</span><span class="p">]))</span>
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<span class="n">file</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
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<span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span></div>
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<div class="viewcode-block" id="data_load"><a class="viewcode-back" href="../data.html#data.data_load">[docs]</a><span class="k">def</span> <span class="nf">data_load</span><span class="p">(</span><span class="n">file_name</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
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<span class="sd">"""Load stored program objects from binary file.</span>
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<span class="sd"> :param file_name: file to load</span>
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<span class="sd"> :type file_name: str</span>
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<span class="sd"> :param verbose: verbose information (default = False)</span>
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<span class="sd"> :type verbose: bool</span>
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<span class="sd"> :returns: loaded data</span>
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<span class="sd"> :rtype: object</span>
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<span class="sd"> """</span>
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<span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
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<span class="nb">print</span><span class="p">(</span><span class="s1">'check if data is available'</span><span class="p">)</span>
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<span class="k">try</span><span class="p">:</span>
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<span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="n">file_name</span><span class="p">,</span> <span class="s1">'rb'</span><span class="p">)</span> <span class="k">as</span> <span class="nb">input</span><span class="p">:</span>
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<span class="n">object_data</span> <span class="o">=</span> <span class="n">pickle</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span> <span class="c1"># one load for every dump is needed to load all the data</span>
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<span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
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<span class="nb">print</span><span class="p">(</span><span class="s1">'found:'</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">object_data</span><span class="p">)</span>
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<span class="k">except</span> <span class="ne">IOError</span><span class="p">:</span>
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<span class="n">object_data</span> <span class="o">=</span> <span class="kc">None</span>
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<span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
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<span class="nb">print</span><span class="p">(</span><span class="s1">'no saved datas found'</span><span class="p">)</span>
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<span class="k">return</span> <span class="n">object_data</span></div>
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<div class="viewcode-block" id="data_store"><a class="viewcode-back" href="../data.html#data.data_store">[docs]</a><span class="k">def</span> <span class="nf">data_store</span><span class="p">(</span><span class="n">file_name</span><span class="p">,</span> <span class="n">object_data</span><span class="p">):</span>
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<span class="sd">"""Store program objects to binary file.</span>
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<span class="sd"> :param file_name: file to store</span>
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<span class="sd"> :type file_name: str</span>
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<span class="sd"> :param object_data: data to store</span>
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<span class="sd"> :type object_data: object</span>
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<span class="sd"> """</span>
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<span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="n">file_name</span><span class="p">,</span> <span class="s1">'wb'</span><span class="p">)</span> <span class="k">as</span> <span class="n">output</span><span class="p">:</span>
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<span class="n">pickle</span><span class="o">.</span><span class="n">dump</span><span class="p">(</span><span class="n">object_data</span><span class="p">,</span> <span class="n">output</span><span class="p">,</span> <span class="n">pickle</span><span class="o">.</span><span class="n">HIGHEST_PROTOCOL</span><span class="p">)</span> <span class="c1"># every dump needs a load</span></div>
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<div class="viewcode-block" id="main"><a class="viewcode-back" href="../data.html#data.main">[docs]</a><span class="k">def</span> <span class="nf">main</span><span class="p">():</span>
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<span class="sd">"""Main function."""</span>
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<span class="n">file_name</span> <span class="o">=</span> <span class="s2">"slit_test_scan.dat"</span>
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<span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">data_read</span><span class="p">(</span><span class="n">file_name</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">y</span><span class="p">)</span></div>
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</pre></div>
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</div>
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</div>
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</div>
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<div class="sphinxsidebar" role="navigation" aria-label="main navigation">
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<div class="sphinxsidebarwrapper">
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<h1 class="logo"><a href="../index.html">pylib</a></h1>
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<h3>Navigation</h3>
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<div class="relations">
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<h3>Related Topics</h3>
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<ul>
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||||
<li><a href="../index.html">Documentation overview</a><ul>
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||||
<li><a href="index.html">Module code</a><ul>
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</ul></li>
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</ul></li>
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</ul>
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</div>
|
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<div id="searchbox" style="display: none" role="search">
|
||||
<h3>Quick search</h3>
|
||||
<div class="searchformwrapper">
|
||||
<form class="search" action="../search.html" method="get">
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||||
<input type="text" name="q" />
|
||||
<input type="submit" value="Go" />
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</form>
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</div>
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</div>
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<script type="text/javascript">$('#searchbox').show(0);</script>
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</div>
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</div>
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<div class="clearer"></div>
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</div>
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<div class="footer">
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©2019, Daniel Weschke.
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|
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Powered by <a href="http://sphinx-doc.org/">Sphinx 2.0.1</a>
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& <a href="https://github.com/bitprophet/alabaster">Alabaster 0.7.12</a>
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</div>
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</body>
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</html>
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222
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222
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<!DOCTYPE html>
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<html xmlns="http://www.w3.org/1999/xhtml">
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<head>
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<meta charset="utf-8" />
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<title>date — pylib 2019.5.19 documentation</title>
|
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<link rel="stylesheet" href="../_static/alabaster.css" type="text/css" />
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<link rel="stylesheet" href="../_static/pygments.css" type="text/css" />
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<link rel="stylesheet" href="../_static/custom.css" type="text/css" />
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<script type="text/javascript" id="documentation_options" data-url_root="../" src="../_static/documentation_options.js"></script>
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<script type="text/javascript" src="../_static/jquery.js"></script>
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<script type="text/javascript" src="../_static/underscore.js"></script>
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<script type="text/javascript" src="../_static/doctools.js"></script>
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<script async="async" type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-AMS-MML_HTMLorMML"></script>
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<link rel="index" title="Index" href="../genindex.html" />
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<link rel="search" title="Search" href="../search.html" />
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<link rel="stylesheet" href="../_static/custom.css" type="text/css" />
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<meta name="viewport" content="width=device-width, initial-scale=0.9, maximum-scale=0.9" />
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</head><body>
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<div class="document">
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<div class="documentwrapper">
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<div class="bodywrapper">
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<div class="body" role="main">
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<h1>Source code for date</h1><div class="highlight"><pre>
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<span></span><span class="ch">#!/usr/bin/env python</span>
|
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<span class="c1"># -*- coding: utf-8 -*-</span>
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<span class="sd">"""Calculate spacial dates.</span>
|
||||
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<span class="sd">:Date: 2018-01-15</span>
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<span class="sd">.. module:: date</span>
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<span class="sd"> :platform: *nix, Windows</span>
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<span class="sd"> :synopsis: Special dates.</span>
|
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<span class="sd"> </span>
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<span class="sd">.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de></span>
|
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<span class="sd">"""</span>
|
||||
<span class="kn">from</span> <span class="nn">__future__</span> <span class="k">import</span> <span class="n">division</span><span class="p">,</span> <span class="n">print_function</span><span class="p">,</span> <span class="n">unicode_literals</span>
|
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<div class="viewcode-block" id="gaußsche_osterformel"><a class="viewcode-back" href="../date.html#date.gaußsche_osterformel">[docs]</a><span class="k">def</span> <span class="nf">gaußsche_osterformel</span><span class="p">(</span><span class="n">year</span><span class="p">):</span>
|
||||
<span class="sd">"""Gaußsche Osterformel.</span>
|
||||
|
||||
<span class="sd"> :param year: the year to calculate the Easter Sunday</span>
|
||||
<span class="sd"> :type year: int</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> :returns: the day of Easter Sunday as a day in march.</span>
|
||||
<span class="sd"> :rtype: int</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> :ivar X: Das Jahr / year</span>
|
||||
<span class="sd"> :vartype X: int</span>
|
||||
<span class="sd"> :ivar K(X): Die Säkularzahl</span>
|
||||
<span class="sd"> :vartype K(X): int</span>
|
||||
<span class="sd"> :ivar M(X): Die säkulare Mondschaltung</span>
|
||||
<span class="sd"> :vartype M(X): int</span>
|
||||
<span class="sd"> :ivar S(K): Die säkulare Sonnenschaltung</span>
|
||||
<span class="sd"> :vartype S(K): int</span>
|
||||
<span class="sd"> :ivar A(X): Den Mondparameter</span>
|
||||
<span class="sd"> :vartype A(X): int</span>
|
||||
<span class="sd"> :ivar D(A,M): Den Keim für den ersten Vollmond im Frühling</span>
|
||||
<span class="sd"> :vartype D(A,M): int</span>
|
||||
<span class="sd"> :ivar R(D,A): Die kalendarische Korrekturgröße</span>
|
||||
<span class="sd"> :vartype R(D,A): int</span>
|
||||
<span class="sd"> :ivar OG(D,R): Die Ostergrenze</span>
|
||||
<span class="sd"> :vartype OG(D,R): int</span>
|
||||
<span class="sd"> :ivar SZ(X,S): Den ersten Sonntag im März</span>
|
||||
<span class="sd"> :vartype SZ(X,S): int</span>
|
||||
<span class="sd"> :ivar OE(OG,SZ): Die Entfernung des Ostersonntags von der Ostergrenze (Osterentfernung in Tagen)</span>
|
||||
<span class="sd"> :vartype OE(OG,SZ): int</span>
|
||||
<span class="sd"> :ivar OS(OG,OE): Das Datum des Ostersonntags als Märzdatum (32. März = 1. April usw.)</span>
|
||||
<span class="sd"> :vartype OS(OG,OE): int</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> Algorithmus gilt für den Gregorianischen Kalender.</span>
|
||||
|
||||
<span class="sd"> source: https://de.wikipedia.org/wiki/Gau%C3%9Fsche_Osterformel</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">year</span>
|
||||
<span class="n">k</span> <span class="o">=</span> <span class="n">x</span> <span class="o">//</span> <span class="mi">100</span>
|
||||
<span class="n">m</span> <span class="o">=</span> <span class="mi">15</span> <span class="o">+</span> <span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">k</span> <span class="o">+</span> <span class="mi">3</span><span class="p">)</span> <span class="o">//</span> <span class="mi">4</span> <span class="o">-</span> <span class="p">(</span><span class="mi">8</span><span class="o">*</span><span class="n">k</span> <span class="o">+</span> <span class="mi">13</span><span class="p">)</span> <span class="o">//</span> <span class="mi">25</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">-</span> <span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">k</span> <span class="o">+</span> <span class="mi">3</span><span class="p">)</span> <span class="o">//</span> <span class="mi">4</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="n">x</span> <span class="o">%</span> <span class="mi">19</span>
|
||||
<span class="n">d</span> <span class="o">=</span> <span class="p">(</span><span class="mi">19</span><span class="o">*</span><span class="n">a</span> <span class="o">+</span> <span class="n">m</span><span class="p">)</span> <span class="o">%</span> <span class="mi">30</span>
|
||||
<span class="n">r</span> <span class="o">=</span> <span class="p">(</span><span class="n">d</span> <span class="o">+</span> <span class="n">a</span> <span class="o">//</span> <span class="mi">11</span><span class="p">)</span> <span class="o">//</span> <span class="mi">29</span>
|
||||
<span class="n">og</span> <span class="o">=</span> <span class="mi">21</span> <span class="o">+</span> <span class="n">d</span> <span class="o">-</span> <span class="n">r</span>
|
||||
<span class="n">sz</span> <span class="o">=</span> <span class="mi">7</span> <span class="o">-</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span> <span class="o">//</span> <span class="mi">4</span> <span class="o">+</span> <span class="n">s</span><span class="p">)</span> <span class="o">%</span> <span class="mi">7</span>
|
||||
<span class="n">oe</span> <span class="o">=</span> <span class="mi">7</span> <span class="o">-</span> <span class="p">(</span><span class="n">og</span> <span class="o">-</span> <span class="n">sz</span><span class="p">)</span> <span class="o">%</span> <span class="mi">7</span>
|
||||
<span class="n">os</span> <span class="o">=</span> <span class="n">og</span> <span class="o">+</span> <span class="n">oe</span>
|
||||
<span class="k">return</span> <span class="n">os</span></div>
|
||||
|
||||
<div class="viewcode-block" id="easter_sunday"><a class="viewcode-back" href="../date.html#date.easter_sunday">[docs]</a><span class="k">def</span> <span class="nf">easter_sunday</span><span class="p">(</span><span class="n">year</span><span class="p">):</span>
|
||||
<span class="sd">"""Easter Sunday.</span>
|
||||
|
||||
<span class="sd"> :param year: the year to calculate the Easter Sunday</span>
|
||||
<span class="sd"> :type year: int</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> :returns: the day of Easter Sunday</span>
|
||||
<span class="sd"> :rtype: datetime.date"""</span>
|
||||
<span class="kn">import</span> <span class="nn">datetime</span>
|
||||
<span class="n">march</span> <span class="o">=</span> <span class="n">datetime</span><span class="o">.</span><span class="n">date</span><span class="p">(</span><span class="n">year</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">day</span> <span class="o">=</span> <span class="n">march</span> <span class="o">+</span> <span class="n">datetime</span><span class="o">.</span><span class="n">timedelta</span><span class="p">(</span><span class="n">days</span><span class="o">=</span><span class="n">gaußsche_osterformel</span><span class="p">(</span><span class="n">year</span><span class="p">))</span>
|
||||
<span class="k">return</span> <span class="n">day</span></div>
|
||||
|
||||
<div class="viewcode-block" id="easter_friday"><a class="viewcode-back" href="../date.html#date.easter_friday">[docs]</a><span class="k">def</span> <span class="nf">easter_friday</span><span class="p">(</span><span class="n">year</span><span class="p">):</span>
|
||||
<span class="sd">"""Easter Friday.</span>
|
||||
|
||||
<span class="sd"> :param year: the year to calculate the Easter Friday</span>
|
||||
<span class="sd"> :type year: int</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> :returns: the day of Easter Friday</span>
|
||||
<span class="sd"> :rtype: datetime.date"""</span>
|
||||
<span class="kn">import</span> <span class="nn">datetime</span>
|
||||
<span class="n">day</span> <span class="o">=</span> <span class="n">easter_sunday</span><span class="p">(</span><span class="n">year</span><span class="p">)</span> <span class="o">+</span> <span class="n">datetime</span><span class="o">.</span><span class="n">timedelta</span><span class="p">(</span><span class="n">days</span><span class="o">=-</span><span class="mi">2</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">day</span></div>
|
||||
|
||||
<div class="viewcode-block" id="easter_monday"><a class="viewcode-back" href="../date.html#date.easter_monday">[docs]</a><span class="k">def</span> <span class="nf">easter_monday</span><span class="p">(</span><span class="n">year</span><span class="p">):</span>
|
||||
<span class="sd">"""Easter Monday.</span>
|
||||
|
||||
<span class="sd"> :param year: the year to calculate the Easter Monday</span>
|
||||
<span class="sd"> :type year: int</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> :returns: the day of Easter Monday</span>
|
||||
<span class="sd"> :rtype: datetime.date"""</span>
|
||||
<span class="kn">import</span> <span class="nn">datetime</span>
|
||||
<span class="n">day</span> <span class="o">=</span> <span class="n">easter_sunday</span><span class="p">(</span><span class="n">year</span><span class="p">)</span> <span class="o">+</span> <span class="n">datetime</span><span class="o">.</span><span class="n">timedelta</span><span class="p">(</span><span class="n">days</span><span class="o">=+</span><span class="mi">1</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">day</span></div>
|
||||
|
||||
<div class="viewcode-block" id="ascension_of_jesus"><a class="viewcode-back" href="../date.html#date.ascension_of_jesus">[docs]</a><span class="k">def</span> <span class="nf">ascension_of_jesus</span><span class="p">(</span><span class="n">year</span><span class="p">):</span>
|
||||
<span class="sd">"""Ascension of Jesus.</span>
|
||||
|
||||
<span class="sd"> :param year: the year to calculate the ascension of Jesus</span>
|
||||
<span class="sd"> :type year: int</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> :returns: the day of ascension of Jesus</span>
|
||||
<span class="sd"> :rtype: datetime.date"""</span>
|
||||
<span class="kn">import</span> <span class="nn">datetime</span>
|
||||
<span class="n">day</span> <span class="o">=</span> <span class="n">easter_sunday</span><span class="p">(</span><span class="n">year</span><span class="p">)</span> <span class="o">+</span> <span class="n">datetime</span><span class="o">.</span><span class="n">timedelta</span><span class="p">(</span><span class="n">days</span><span class="o">=+</span><span class="mi">39</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">day</span></div>
|
||||
|
||||
<div class="viewcode-block" id="pentecost"><a class="viewcode-back" href="../date.html#date.pentecost">[docs]</a><span class="k">def</span> <span class="nf">pentecost</span><span class="p">(</span><span class="n">year</span><span class="p">):</span>
|
||||
<span class="sd">"""Pentecost.</span>
|
||||
|
||||
<span class="sd"> :param year: the year to calculate the Pentecost</span>
|
||||
<span class="sd"> :type year: int</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> :returns: the day of Pentecost</span>
|
||||
<span class="sd"> :rtype: datetime.date"""</span>
|
||||
<span class="kn">import</span> <span class="nn">datetime</span>
|
||||
<span class="n">day</span> <span class="o">=</span> <span class="n">easter_sunday</span><span class="p">(</span><span class="n">year</span><span class="p">)</span> <span class="o">+</span> <span class="n">datetime</span><span class="o">.</span><span class="n">timedelta</span><span class="p">(</span><span class="n">days</span><span class="o">=+</span><span class="mi">49</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">day</span></div>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
<div class="sphinxsidebar" role="navigation" aria-label="main navigation">
|
||||
<div class="sphinxsidebarwrapper">
|
||||
<h1 class="logo"><a href="../index.html">pylib</a></h1>
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
<h3>Navigation</h3>
|
||||
|
||||
<div class="relations">
|
||||
<h3>Related Topics</h3>
|
||||
<ul>
|
||||
<li><a href="../index.html">Documentation overview</a><ul>
|
||||
<li><a href="index.html">Module code</a><ul>
|
||||
</ul></li>
|
||||
</ul></li>
|
||||
</ul>
|
||||
</div>
|
||||
<div id="searchbox" style="display: none" role="search">
|
||||
<h3>Quick search</h3>
|
||||
<div class="searchformwrapper">
|
||||
<form class="search" action="../search.html" method="get">
|
||||
<input type="text" name="q" />
|
||||
<input type="submit" value="Go" />
|
||||
</form>
|
||||
</div>
|
||||
</div>
|
||||
<script type="text/javascript">$('#searchbox').show(0);</script>
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
</div>
|
||||
</div>
|
||||
<div class="clearer"></div>
|
||||
</div>
|
||||
<div class="footer">
|
||||
©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>
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
306
docs/build/html/_modules/geometry.html
vendored
Normal file
306
docs/build/html/_modules/geometry.html
vendored
Normal file
@@ -0,0 +1,306 @@
|
||||
|
||||
<!DOCTYPE html>
|
||||
|
||||
<html xmlns="http://www.w3.org/1999/xhtml">
|
||||
<head>
|
||||
<meta charset="utf-8" />
|
||||
<title>geometry — pylib 2019.5.19 documentation</title>
|
||||
<link rel="stylesheet" href="../_static/alabaster.css" type="text/css" />
|
||||
<link rel="stylesheet" href="../_static/pygments.css" type="text/css" />
|
||||
<link rel="stylesheet" href="../_static/custom.css" type="text/css" />
|
||||
<script type="text/javascript" id="documentation_options" data-url_root="../" src="../_static/documentation_options.js"></script>
|
||||
<script type="text/javascript" src="../_static/jquery.js"></script>
|
||||
<script type="text/javascript" src="../_static/underscore.js"></script>
|
||||
<script type="text/javascript" src="../_static/doctools.js"></script>
|
||||
<script type="text/javascript" src="../_static/language_data.js"></script>
|
||||
<script async="async" type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-AMS-MML_HTMLorMML"></script>
|
||||
<link rel="index" title="Index" href="../genindex.html" />
|
||||
<link rel="search" title="Search" href="../search.html" />
|
||||
|
||||
<link rel="stylesheet" href="../_static/custom.css" type="text/css" />
|
||||
|
||||
|
||||
<meta name="viewport" content="width=device-width, initial-scale=0.9, maximum-scale=0.9" />
|
||||
|
||||
</head><body>
|
||||
|
||||
|
||||
<div class="document">
|
||||
<div class="documentwrapper">
|
||||
<div class="bodywrapper">
|
||||
|
||||
|
||||
<div class="body" role="main">
|
||||
|
||||
<h1>Source code for geometry</h1><div class="highlight"><pre>
|
||||
<span></span><span class="ch">#!/usr/bin/env python</span>
|
||||
<span class="c1"># -*- coding: utf-8 -*-</span>
|
||||
<span class="sd">"""2D geometry objects.</span>
|
||||
|
||||
<span class="sd">:Date: 2019-03-21</span>
|
||||
|
||||
<span class="sd">.. module:: geometry</span>
|
||||
<span class="sd"> :platform: *nix, Windows</span>
|
||||
<span class="sd"> :synopsis: Geometry objects.</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd">.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de></span>
|
||||
<span class="sd">"""</span>
|
||||
<span class="kn">import</span> <span class="nn">math</span>
|
||||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||||
|
||||
|
||||
<div class="viewcode-block" id="translate"><a class="viewcode-back" href="../geometry.html#geometry.translate">[docs]</a><span class="k">def</span> <span class="nf">translate</span><span class="p">(</span><span class="n">vec</span><span class="p">,</span> <span class="o">*</span><span class="n">pts</span><span class="p">):</span>
|
||||
<span class="sd">"""Translate a point or polygon by a given vector.</span>
|
||||
|
||||
<span class="sd"> :param vec: translation vector</span>
|
||||
<span class="sd"> :type vec: tuple</span>
|
||||
<span class="sd"> :param `*pts`: points to translate</span>
|
||||
|
||||
<span class="sd"> :returns: (point_x, point_y) or (point1, point2, ...)</span>
|
||||
<span class="sd"> :rtype: tuple</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">vx</span><span class="p">,</span> <span class="n">vy</span> <span class="o">=</span> <span class="n">vec</span>
|
||||
<span class="k">return</span> <span class="nb">tuple</span><span class="p">([(</span><span class="n">x</span><span class="o">+</span><span class="n">vx</span><span class="p">,</span> <span class="n">y</span><span class="o">+</span><span class="n">vy</span><span class="p">)</span> <span class="k">for</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span> <span class="ow">in</span> <span class="n">pts</span><span class="p">])</span></div>
|
||||
|
||||
|
||||
<div class="viewcode-block" id="rotate"><a class="viewcode-back" href="../geometry.html#geometry.rotate">[docs]</a><span class="k">def</span> <span class="nf">rotate</span><span class="p">(</span><span class="n">origin</span><span class="p">,</span> <span class="n">angle</span><span class="p">,</span> <span class="o">*</span><span class="n">pts</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
|
||||
<span class="sd">"""Rotate a point or polygon counterclockwise by a given angle around a given</span>
|
||||
<span class="sd"> origin. The angle should be given in radians.</span>
|
||||
|
||||
<span class="sd"> :param origin: the center of rotation</span>
|
||||
<span class="sd"> :type origin: tuple</span>
|
||||
<span class="sd"> :param angle: the rotation angle</span>
|
||||
<span class="sd"> :type angle: int or float</span>
|
||||
<span class="sd"> :param `*pts`: points to rotate</span>
|
||||
<span class="sd"> :param `**kwargs`: options</span>
|
||||
|
||||
<span class="sd"> :returns: (point_x, point_y) or (point1, point2, ...)</span>
|
||||
<span class="sd"> :rtype: tuple</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">ox</span><span class="p">,</span> <span class="n">oy</span> <span class="o">=</span> <span class="n">origin</span>
|
||||
|
||||
<span class="c1"># add first point to the end</span>
|
||||
<span class="k">if</span> <span class="n">kwargs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="ow">and</span> <span class="s2">"closed"</span> <span class="ow">in</span> <span class="n">kwargs</span> <span class="ow">and</span> <span class="n">kwargs</span><span class="p">[</span><span class="s2">"closed"</span><span class="p">]</span> <span class="ow">is</span> <span class="kc">True</span><span class="p">:</span>
|
||||
<span class="n">pts</span> <span class="o">+=</span> <span class="p">(</span><span class="n">pts</span><span class="p">[</span><span class="mi">0</span><span class="p">],)</span>
|
||||
|
||||
<span class="n">result</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">([(</span><span class="n">ox</span> <span class="o">+</span> <span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">px</span> <span class="o">-</span> <span class="n">ox</span><span class="p">)</span> <span class="o">-</span> <span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">py</span> <span class="o">-</span> <span class="n">oy</span><span class="p">),</span>
|
||||
<span class="n">oy</span> <span class="o">+</span> <span class="n">math</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">px</span> <span class="o">-</span> <span class="n">ox</span><span class="p">)</span> <span class="o">+</span> <span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">py</span> <span class="o">-</span> <span class="n">oy</span><span class="p">))</span>
|
||||
<span class="k">for</span> <span class="p">(</span><span class="n">px</span><span class="p">,</span> <span class="n">py</span><span class="p">)</span> <span class="ow">in</span> <span class="n">pts</span><span class="p">])</span>
|
||||
|
||||
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">pts</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="n">result</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">result</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="k">return</span> <span class="n">result</span></div>
|
||||
|
||||
|
||||
<div class="viewcode-block" id="rotate_deg"><a class="viewcode-back" href="../geometry.html#geometry.rotate_deg">[docs]</a><span class="k">def</span> <span class="nf">rotate_deg</span><span class="p">(</span><span class="n">origin</span><span class="p">,</span> <span class="n">angle</span><span class="p">,</span> <span class="o">*</span><span class="n">pts</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
|
||||
<span class="sd">"""Rotate a point or polygon counterclockwise by a given angle around a given</span>
|
||||
<span class="sd"> origin. The angle should be given in degrees.</span>
|
||||
|
||||
<span class="sd"> :param origin: the center of rotation</span>
|
||||
<span class="sd"> :type origin: tuple</span>
|
||||
<span class="sd"> :param angle: the rotation angle</span>
|
||||
<span class="sd"> :type angle: int or float</span>
|
||||
<span class="sd"> :param `*pts`: points to rotate</span>
|
||||
<span class="sd"> :param `**kwargs`: options</span>
|
||||
|
||||
<span class="sd"> :returns: (point_x, point_y) or (point1, point2, ...)</span>
|
||||
<span class="sd"> :rtype: tuple</span>
|
||||
|
||||
<span class="sd"> .. seealso::</span>
|
||||
<span class="sd"> :meth:`rotate`</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="k">return</span> <span class="n">rotate</span><span class="p">(</span><span class="n">origin</span><span class="p">,</span> <span class="n">angle</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">180</span><span class="p">,</span> <span class="o">*</span><span class="n">pts</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
|
||||
|
||||
|
||||
<div class="viewcode-block" id="rectangle"><a class="viewcode-back" href="../geometry.html#geometry.rectangle">[docs]</a><span class="k">def</span> <span class="nf">rectangle</span><span class="p">(</span><span class="n">width</span><span class="p">,</span> <span class="n">height</span><span class="p">):</span>
|
||||
<span class="sd">"""\</span>
|
||||
<span class="sd"> :param width: the width of the rectangle</span>
|
||||
<span class="sd"> :type width: int or float</span>
|
||||
<span class="sd"> :param height: the height of the rectangle</span>
|
||||
<span class="sd"> :type height: int or float</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> :returns: (point1, point2, point3, point4)</span>
|
||||
<span class="sd"> :rtype: tuple</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">pt1</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">width</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="n">height</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
|
||||
<span class="n">pt2</span> <span class="o">=</span> <span class="p">(</span><span class="n">width</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="n">height</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
|
||||
<span class="n">pt3</span> <span class="o">=</span> <span class="p">(</span><span class="n">width</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">height</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
|
||||
<span class="n">pt4</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">width</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">height</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">pt1</span><span class="p">,</span> <span class="n">pt2</span><span class="p">,</span> <span class="n">pt3</span><span class="p">,</span> <span class="n">pt4</span><span class="p">,</span> <span class="n">pt1</span></div>
|
||||
|
||||
|
||||
<div class="viewcode-block" id="square"><a class="viewcode-back" href="../geometry.html#geometry.square">[docs]</a><span class="k">def</span> <span class="nf">square</span><span class="p">(</span><span class="n">width</span><span class="p">):</span>
|
||||
<span class="sd">"""\</span>
|
||||
<span class="sd"> :param width: the edge size of the square</span>
|
||||
<span class="sd"> :type width: int or float</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> :returns: (point1, point2, point3, point4)</span>
|
||||
<span class="sd"> :rtype: tuple</span>
|
||||
|
||||
<span class="sd"> .. seealso::</span>
|
||||
<span class="sd"> :meth:`rectangle`</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="k">return</span> <span class="n">rectangle</span><span class="p">(</span><span class="n">width</span><span class="p">,</span> <span class="n">width</span><span class="p">)</span></div>
|
||||
|
||||
|
||||
<span class="c1">#</span>
|
||||
<span class="c1"># matplotlib format, return lists for x and y</span>
|
||||
<span class="c1">#</span>
|
||||
|
||||
<div class="viewcode-block" id="points"><a class="viewcode-back" href="../geometry.html#geometry.points">[docs]</a><span class="k">def</span> <span class="nf">points</span><span class="p">(</span><span class="o">*</span><span class="n">pts</span><span class="p">):</span>
|
||||
<span class="sd">"""\</span>
|
||||
<span class="sd"> :param `*pts`: points to rearrange</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> :returns: ((point1_x, point2_x), (point1_y, point2_y), ...)</span>
|
||||
<span class="sd"> :rtype: tuple</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="k">return</span> <span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">pts</span><span class="p">)</span></div>
|
||||
|
||||
|
||||
<div class="viewcode-block" id="line"><a class="viewcode-back" href="../geometry.html#geometry.line">[docs]</a><span class="k">def</span> <span class="nf">line</span><span class="p">(</span><span class="n">point1</span><span class="p">,</span> <span class="n">point2</span><span class="p">,</span> <span class="n">samples</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
|
||||
<span class="sd">"""\</span>
|
||||
<span class="sd"> .. math::</span>
|
||||
<span class="sd"> y = \\frac{y_2-y_1}{x_2-x_1}(x-x_1) + y_1</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> :param point1: one end point</span>
|
||||
<span class="sd"> :type point1: tuple</span>
|
||||
<span class="sd"> :param point2: other end point</span>
|
||||
<span class="sd"> :type point2: tuple</span>
|
||||
<span class="sd"> :param samples: number of sampling points</span>
|
||||
<span class="sd"> :type samples: int</span>
|
||||
|
||||
<span class="sd"> :returns: ((point1_x, point2_x), (points1_y, point2_y)) or</span>
|
||||
<span class="sd"> ([sample_point1_x, sample_point2_x, ...],</span>
|
||||
<span class="sd"> [sample_points1_y, sample_point2_y, ...])</span>
|
||||
<span class="sd"> :rtype: tuple</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">p1x</span><span class="p">,</span> <span class="n">p1y</span> <span class="o">=</span> <span class="n">point1</span>
|
||||
<span class="n">p2x</span><span class="p">,</span> <span class="n">p2y</span> <span class="o">=</span> <span class="n">point2</span>
|
||||
|
||||
<span class="n">denominator</span> <span class="o">=</span> <span class="p">(</span><span class="n">p1x</span> <span class="o">-</span> <span class="n">p2x</span><span class="p">)</span>
|
||||
|
||||
<span class="k">if</span> <span class="n">samples</span> <span class="o">></span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">denominator</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">p1x</span><span class="p">,</span> <span class="n">p2x</span><span class="p">)</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="p">(</span><span class="n">p1y</span> <span class="o">-</span> <span class="n">p2y</span><span class="p">)</span> <span class="o">/</span> <span class="n">denominator</span>
|
||||
<span class="n">b</span> <span class="o">=</span> <span class="p">(</span><span class="n">p1x</span><span class="o">*</span><span class="n">p2y</span> <span class="o">-</span> <span class="n">p2x</span><span class="o">*</span><span class="n">p1y</span><span class="p">)</span> <span class="o">/</span> <span class="n">denominator</span>
|
||||
<span class="n">y</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">b</span>
|
||||
<span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
|
||||
<span class="k">return</span> <span class="p">(</span><span class="n">p1x</span><span class="p">,</span> <span class="n">p2x</span><span class="p">),</span> <span class="p">(</span><span class="n">p1y</span><span class="p">,</span> <span class="n">p2y</span><span class="p">)</span> <span class="c1"># matplotlib format</span></div>
|
||||
|
||||
|
||||
<div class="viewcode-block" id="cubic"><a class="viewcode-back" href="../geometry.html#geometry.cubic">[docs]</a><span class="k">def</span> <span class="nf">cubic</span><span class="p">(</span><span class="n">point1</span><span class="p">,</span> <span class="n">angle1</span><span class="p">,</span> <span class="n">point2</span><span class="p">,</span> <span class="n">angle2</span><span class="p">,</span> <span class="n">samples</span><span class="o">=</span><span class="mi">50</span><span class="p">):</span>
|
||||
<span class="sd">"""\</span>
|
||||
<span class="sd"> :param point1: one end point</span>
|
||||
<span class="sd"> :type point1: tuple</span>
|
||||
<span class="sd"> :param angle1: the slope at the one end point</span>
|
||||
<span class="sd"> :type angle1: int or float</span>
|
||||
<span class="sd"> :param point2: other end point</span>
|
||||
<span class="sd"> :type point2: tuple</span>
|
||||
<span class="sd"> :param angle2: the slope at the other end point</span>
|
||||
<span class="sd"> :type angle2: int or float</span>
|
||||
<span class="sd"> :param samples: number of sampling points</span>
|
||||
<span class="sd"> :type samples: int</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> :returns: ([sample_point1_x, sample_point2_x, ...],</span>
|
||||
<span class="sd"> [sample_points1_y, sample_point2_y, ...])</span>
|
||||
<span class="sd"> :rtype: tuple</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">p1x</span><span class="p">,</span> <span class="n">p1y</span> <span class="o">=</span> <span class="n">point1</span>
|
||||
<span class="n">p2x</span><span class="p">,</span> <span class="n">p2y</span> <span class="o">=</span> <span class="n">point2</span>
|
||||
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">p1x</span><span class="p">,</span> <span class="n">p2x</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="n">samples</span><span class="p">)</span>
|
||||
|
||||
<span class="n">p1ys</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">tan</span><span class="p">(</span><span class="n">angle1</span><span class="p">)</span>
|
||||
<span class="n">p2ys</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">tan</span><span class="p">(</span><span class="n">angle2</span><span class="p">)</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="p">(</span><span class="n">p1x</span><span class="o">*</span><span class="n">p1ys</span> <span class="o">+</span> <span class="n">p1x</span><span class="o">*</span><span class="n">p2ys</span> <span class="o">-</span> <span class="n">p2x</span><span class="o">*</span><span class="n">p1ys</span> <span class="o">-</span> <span class="n">p2x</span><span class="o">*</span><span class="n">p2ys</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">p1y</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">p2y</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">p1x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">p1x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p2x</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">p1x</span><span class="o">*</span><span class="n">p2x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">p2x</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span>
|
||||
<span class="n">b</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span> <span class="n">p1x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p1ys</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">p1x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p2ys</span> <span class="o">-</span> <span class="n">p1x</span><span class="o">*</span><span class="n">p2x</span><span class="o">*</span><span class="n">p1ys</span> <span class="o">+</span> <span class="n">p1x</span><span class="o">*</span><span class="n">p2x</span><span class="o">*</span><span class="n">p2ys</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">p1x</span><span class="o">*</span><span class="n">p1y</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">p1x</span><span class="o">*</span><span class="n">p2y</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">p2x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p1ys</span> <span class="o">+</span> <span class="n">p2x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p2ys</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">p2x</span><span class="o">*</span><span class="n">p1y</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">p2x</span><span class="o">*</span><span class="n">p2y</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">p1x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">p1x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p2x</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">p1x</span><span class="o">*</span><span class="n">p2x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">p2x</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span>
|
||||
<span class="n">c</span> <span class="o">=</span> <span class="p">(</span><span class="n">p1x</span><span class="o">**</span><span class="mi">3</span><span class="o">*</span><span class="n">p2ys</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">p1x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p2x</span><span class="o">*</span><span class="n">p1ys</span> <span class="o">+</span> <span class="n">p1x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p2x</span><span class="o">*</span><span class="n">p2ys</span> <span class="o">-</span> <span class="n">p1x</span><span class="o">*</span><span class="n">p2x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p1ys</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">p1x</span><span class="o">*</span><span class="n">p2x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p2ys</span> <span class="o">-</span> <span class="mi">6</span><span class="o">*</span><span class="n">p1x</span><span class="o">*</span><span class="n">p2x</span><span class="o">*</span><span class="n">p1y</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">p1x</span><span class="o">*</span><span class="n">p2x</span><span class="o">*</span><span class="n">p2y</span> <span class="o">-</span> <span class="n">p2x</span><span class="o">**</span><span class="mi">3</span><span class="o">*</span><span class="n">p1ys</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">p1x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">p1x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p2x</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">p1x</span><span class="o">*</span><span class="n">p2x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">p2x</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span>
|
||||
<span class="n">d</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span> <span class="n">p1x</span><span class="o">**</span><span class="mi">3</span><span class="o">*</span><span class="n">p2x</span><span class="o">*</span><span class="n">p2ys</span> <span class="o">+</span> <span class="n">p1x</span><span class="o">**</span><span class="mi">3</span><span class="o">*</span><span class="n">p2y</span> <span class="o">-</span> <span class="n">p1x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p2x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p1ys</span> <span class="o">+</span> <span class="n">p1x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p2x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p2ys</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">p1x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p2x</span><span class="o">*</span><span class="n">p2y</span> <span class="o">+</span> <span class="n">p1x</span><span class="o">*</span><span class="n">p2x</span><span class="o">**</span><span class="mi">3</span><span class="o">*</span><span class="n">p1ys</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">p1x</span><span class="o">*</span><span class="n">p2x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p1y</span> <span class="o">-</span> <span class="n">p2x</span><span class="o">**</span><span class="mi">3</span><span class="o">*</span><span class="n">p1y</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">p1x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">p1x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">p2x</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">p1x</span><span class="o">*</span><span class="n">p2x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">p2x</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span>
|
||||
<span class="n">y</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">d</span>
|
||||
<span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span></div>
|
||||
|
||||
|
||||
<div class="viewcode-block" id="cubic_deg"><a class="viewcode-back" href="../geometry.html#geometry.cubic_deg">[docs]</a><span class="k">def</span> <span class="nf">cubic_deg</span><span class="p">(</span><span class="n">point1</span><span class="p">,</span> <span class="n">angle1</span><span class="p">,</span> <span class="n">point2</span><span class="p">,</span> <span class="n">angle2</span><span class="p">):</span>
|
||||
<span class="sd">"""\</span>
|
||||
<span class="sd"> :param point1: one end point</span>
|
||||
<span class="sd"> :type point1: tuple</span>
|
||||
<span class="sd"> :param angle1: the slope at the one end point</span>
|
||||
<span class="sd"> :type angle1: int or float</span>
|
||||
<span class="sd"> :param point2: other end point</span>
|
||||
<span class="sd"> :type point2: tuple</span>
|
||||
<span class="sd"> :param angle2: the slope at the other end point</span>
|
||||
<span class="sd"> :type angle2: int or float</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> :returns: ([sample_point1_x, sample_point2_x, ...],</span>
|
||||
<span class="sd"> [sample_points1_y, sample_point2_y, ...])</span>
|
||||
<span class="sd"> :rtype: tuple</span>
|
||||
|
||||
<span class="sd"> .. seealso::</span>
|
||||
<span class="sd"> :meth:`cubic`</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="k">return</span> <span class="n">cubic</span><span class="p">(</span><span class="n">point1</span><span class="p">,</span> <span class="n">angle1</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">180</span><span class="p">,</span> <span class="n">point2</span><span class="p">,</span> <span class="n">angle2</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">180</span><span class="p">)</span></div>
|
||||
</pre></div>
|
||||
|
||||
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|
||||
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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Powered by <a href="http://sphinx-doc.org/">Sphinx 2.0.1</a>
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104
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|
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|
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<!DOCTYPE html>
|
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|
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|
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|
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<meta charset="utf-8" />
|
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<title>Overview: module code — pylib 2019.5.19 documentation</title>
|
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|
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|
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|
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<li><a href="date.html">date</a></li>
|
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<li><a href="geometry.html">geometry</a></li>
|
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<li><a href="numerical/fit.html">numerical.fit</a></li>
|
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<li><a href="numerical/integration.html">numerical.integration</a></li>
|
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<li><a href="numerical/ode.html">numerical.ode</a></li>
|
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<li><a href="numerical/ode_model.html">numerical.ode_model</a></li>
|
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<h3>Navigation</h3>
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<h3>Related Topics</h3>
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<div id="searchbox" style="display: none" role="search">
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<h3>Quick search</h3>
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<form class="search" action="../search.html" method="get">
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<input type="submit" value="Go" />
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|
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<div class="footer">
|
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©2019, Daniel Weschke.
|
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|
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|
|
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Powered by <a href="http://sphinx-doc.org/">Sphinx 2.0.1</a>
|
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& <a href="https://github.com/bitprophet/alabaster">Alabaster 0.7.12</a>
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|
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|
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|
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194
docs/build/html/_modules/numerical/fit.html
vendored
Normal file
194
docs/build/html/_modules/numerical/fit.html
vendored
Normal file
@@ -0,0 +1,194 @@
|
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|
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<!DOCTYPE html>
|
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|
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<html xmlns="http://www.w3.org/1999/xhtml">
|
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|
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<meta charset="utf-8" />
|
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<title>numerical.fit — pylib 2019.5.19 documentation</title>
|
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|
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<div class="body" role="main">
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|
||||
<h1>Source code for numerical.fit</h1><div class="highlight"><pre>
|
||||
<span></span><span class="ch">#!/usr/bin/env python</span>
|
||||
<span class="c1"># -*- coding: utf-8 -*-</span>
|
||||
<span class="sd">"""Function and approximation.</span>
|
||||
|
||||
<span class="sd">.. module:: fit</span>
|
||||
<span class="sd"> :platform: *nix, Windows</span>
|
||||
<span class="sd"> :synopsis: Function and approximation.</span>
|
||||
|
||||
<span class="sd">.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de></span>
|
||||
<span class="sd">"""</span>
|
||||
<span class="kn">from</span> <span class="nn">__future__</span> <span class="k">import</span> <span class="n">print_function</span>
|
||||
<span class="kn">from</span> <span class="nn">pylab</span> <span class="k">import</span> <span class="n">array</span><span class="p">,</span> <span class="n">argmax</span><span class="p">,</span> <span class="n">gradient</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">linspace</span>
|
||||
<span class="kn">from</span> <span class="nn">scipy.optimize</span> <span class="k">import</span> <span class="n">curve_fit</span>
|
||||
|
||||
<div class="viewcode-block" id="gauss"><a class="viewcode-back" href="../../numerical.html#numerical.fit.gauss">[docs]</a><span class="k">def</span> <span class="nf">gauss</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">):</span>
|
||||
<span class="sd">"""Gauss distribution function.</span>
|
||||
|
||||
<span class="sd"> .. math::</span>
|
||||
<span class="sd"> f(x)=ae^{-(x-b)^{2}/(2c^{2})}</span>
|
||||
|
||||
<span class="sd"> :param x: positions where the gauss function will be calculated</span>
|
||||
<span class="sd"> :type x: int or float or list or numpy.ndarray</span>
|
||||
<span class="sd"> :param p: gauss parameters [a, b, c, d]:</span>
|
||||
|
||||
<span class="sd"> * a -- amplitude (:math:`\int y \\,\\mathrm{d}x=1 \Leftrightarrow a=1/(c\\sqrt{2\\pi})` )</span>
|
||||
<span class="sd"> * b -- expected value :math:`\\mu` (position of maximum, default = 0)</span>
|
||||
<span class="sd"> * c -- standard deviation :math:`\\sigma` (variance :math:`\\sigma^2=c^2`)</span>
|
||||
<span class="sd"> * d -- vertical offset (default = 0)</span>
|
||||
<span class="sd"> :type p: list</span>
|
||||
|
||||
<span class="sd"> :returns: gauss values at given positions x</span>
|
||||
<span class="sd"> :rtype: numpy.ndarray</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="c1"># cast e. g. list to numpy array</span>
|
||||
<span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">p</span>
|
||||
<span class="k">return</span> <span class="n">a</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mf">2.</span><span class="o">/</span><span class="p">(</span><span class="mf">2.</span> <span class="o">*</span> <span class="n">c</span><span class="o">**</span><span class="mf">2.</span><span class="p">))</span> <span class="o">+</span> <span class="n">d</span></div>
|
||||
|
||||
<div class="viewcode-block" id="gauss_fit"><a class="viewcode-back" href="../../numerical.html#numerical.fit.gauss_fit">[docs]</a><span class="k">def</span> <span class="nf">gauss_fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">e</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">x_fit</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="sd">"""Fit Gauss distribution function to data.</span>
|
||||
|
||||
<span class="sd"> :param x: positions</span>
|
||||
<span class="sd"> :type x: int or float or list or numpy.ndarray</span>
|
||||
<span class="sd"> :param y: values</span>
|
||||
<span class="sd"> :type y: int or float or list or numpy.ndarray</span>
|
||||
<span class="sd"> :param e: error values (default = None)</span>
|
||||
<span class="sd"> :type e: int or float or list or numpy.ndarray</span>
|
||||
<span class="sd"> :param x_fit: positions of fitted function (default = None, if None then x</span>
|
||||
<span class="sd"> is used)</span>
|
||||
<span class="sd"> :type x_fit: int or float or list or numpy.ndarray</span>
|
||||
<span class="sd"> :param verbose: verbose information (default = False)</span>
|
||||
<span class="sd"> :type verbose: bool</span>
|
||||
|
||||
<span class="sd"> :returns:</span>
|
||||
<span class="sd"> * numpy.ndarray -- fitted values (y_fit)</span>
|
||||
<span class="sd"> * numpy.ndarray -- parameters of gauss distribution function (popt:</span>
|
||||
<span class="sd"> amplitude a, expected value :math:`\\mu`, standard deviation</span>
|
||||
<span class="sd"> :math:`\\sigma`, vertical offset d)</span>
|
||||
<span class="sd"> * numpy.float64 -- full width at half maximum (FWHM)</span>
|
||||
<span class="sd"> :rtype: tuple</span>
|
||||
|
||||
<span class="sd"> .. seealso::</span>
|
||||
<span class="sd"> :meth:`gauss`</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="c1"># cast e. g. list to numpy array</span>
|
||||
<span class="n">y</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="n">y</span><span class="p">)</span> <span class="c1"># cast e. g. list to numpy array</span>
|
||||
<span class="n">y_max</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
|
||||
<span class="n">y_max_pos</span> <span class="o">=</span> <span class="n">argmax</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
|
||||
<span class="n">x_y_max</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">y_max_pos</span><span class="p">]</span>
|
||||
|
||||
<span class="c1"># starting parameter</span>
|
||||
<span class="n">p0</span> <span class="o">=</span> <span class="p">[</span><span class="n">y_max</span><span class="p">,</span> <span class="n">x_y_max</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="p">[</span><span class="mi">0</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="s1">'p0:'</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s1">' '</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">p0</span><span class="p">)</span>
|
||||
<span class="k">if</span> <span class="n">e</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
|
||||
<span class="n">popt</span><span class="p">,</span> <span class="n">pcov</span> <span class="o">=</span> <span class="n">curve_fit</span><span class="p">(</span><span class="n">gauss</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">p0</span><span class="o">=</span><span class="n">p0</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="n">e</span><span class="p">)</span>
|
||||
<span class="k">else</span><span class="p">:</span>
|
||||
<span class="n">popt</span><span class="p">,</span> <span class="n">pcov</span> <span class="o">=</span> <span class="n">curve_fit</span><span class="p">(</span><span class="n">gauss</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">p0</span><span class="o">=</span><span class="n">p0</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="s1">'popt:'</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s1">' '</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">popt</span><span class="p">)</span>
|
||||
<span class="c1">#print(pcov)</span>
|
||||
|
||||
<span class="n">FWHM</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">*</span><span class="n">popt</span><span class="p">[</span><span class="mi">2</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="s1">'FWHM'</span><span class="p">,</span> <span class="n">FWHM</span><span class="p">)</span>
|
||||
|
||||
<span class="k">if</span> <span class="n">x_fit</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
|
||||
<span class="n">x_fit</span> <span class="o">=</span> <span class="n">x</span>
|
||||
<span class="n">y_fit</span> <span class="o">=</span> <span class="n">gauss</span><span class="p">(</span><span class="n">x_fit</span><span class="p">,</span> <span class="o">*</span><span class="n">popt</span><span class="p">)</span>
|
||||
|
||||
<span class="k">return</span> <span class="n">y_fit</span><span class="p">,</span> <span class="n">popt</span><span class="p">,</span> <span class="n">FWHM</span></div>
|
||||
|
||||
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s2">"__main__"</span><span class="p">:</span>
|
||||
<span class="kc">True</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
<div class="sphinxsidebar" role="navigation" aria-label="main navigation">
|
||||
<div class="sphinxsidebarwrapper">
|
||||
<h1 class="logo"><a href="../../index.html">pylib</a></h1>
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
<h3>Navigation</h3>
|
||||
|
||||
<div class="relations">
|
||||
<h3>Related Topics</h3>
|
||||
<ul>
|
||||
<li><a href="../../index.html">Documentation overview</a><ul>
|
||||
<li><a href="../index.html">Module code</a><ul>
|
||||
</ul></li>
|
||||
</ul></li>
|
||||
</ul>
|
||||
</div>
|
||||
<div id="searchbox" style="display: none" role="search">
|
||||
<h3>Quick search</h3>
|
||||
<div class="searchformwrapper">
|
||||
<form class="search" action="../../search.html" method="get">
|
||||
<input type="text" name="q" />
|
||||
<input type="submit" value="Go" />
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</form>
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||||
</div>
|
||||
</div>
|
||||
<script type="text/javascript">$('#searchbox').show(0);</script>
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
</div>
|
||||
</div>
|
||||
<div class="clearer"></div>
|
||||
</div>
|
||||
<div class="footer">
|
||||
©2019, Daniel Weschke.
|
||||
|
||||
|
|
||||
Powered by <a href="http://sphinx-doc.org/">Sphinx 2.0.1</a>
|
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& <a href="https://github.com/bitprophet/alabaster">Alabaster 0.7.12</a>
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|
||||
</div>
|
||||
|
||||
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
250
docs/build/html/_modules/numerical/integration.html
vendored
Normal file
250
docs/build/html/_modules/numerical/integration.html
vendored
Normal file
@@ -0,0 +1,250 @@
|
||||
|
||||
<!DOCTYPE html>
|
||||
|
||||
<html xmlns="http://www.w3.org/1999/xhtml">
|
||||
<head>
|
||||
<meta charset="utf-8" />
|
||||
<title>numerical.integration — pylib 2019.5.19 documentation</title>
|
||||
<link rel="stylesheet" href="../../_static/alabaster.css" type="text/css" />
|
||||
<link rel="stylesheet" href="../../_static/pygments.css" type="text/css" />
|
||||
<link rel="stylesheet" href="../../_static/custom.css" type="text/css" />
|
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<script type="text/javascript" id="documentation_options" data-url_root="../../" src="../../_static/documentation_options.js"></script>
|
||||
<script type="text/javascript" src="../../_static/jquery.js"></script>
|
||||
<script type="text/javascript" src="../../_static/underscore.js"></script>
|
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<script type="text/javascript" src="../../_static/doctools.js"></script>
|
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<script type="text/javascript" src="../../_static/language_data.js"></script>
|
||||
<script async="async" type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-AMS-MML_HTMLorMML"></script>
|
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<link rel="index" title="Index" href="../../genindex.html" />
|
||||
<link rel="search" title="Search" href="../../search.html" />
|
||||
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<link rel="stylesheet" href="../../_static/custom.css" type="text/css" />
|
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|
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|
||||
<meta name="viewport" content="width=device-width, initial-scale=0.9, maximum-scale=0.9" />
|
||||
|
||||
</head><body>
|
||||
|
||||
|
||||
<div class="document">
|
||||
<div class="documentwrapper">
|
||||
<div class="bodywrapper">
|
||||
|
||||
|
||||
<div class="body" role="main">
|
||||
|
||||
<h1>Source code for numerical.integration</h1><div class="highlight"><pre>
|
||||
<span></span><span class="c1"># -*- coding: utf-8 -*-</span>
|
||||
<span class="sd">"""Numerical integration, numerical quadrature.</span>
|
||||
|
||||
<span class="sd">de: numerische Integration, numerische Quadratur.</span>
|
||||
|
||||
<span class="sd">:Date: 2015-10-15</span>
|
||||
|
||||
<span class="sd">.. module:: integration</span>
|
||||
<span class="sd"> :platform: *nix, Windows</span>
|
||||
<span class="sd"> :synopsis: Numerical integration.</span>
|
||||
|
||||
<span class="sd">.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de></span>
|
||||
<span class="sd">"""</span>
|
||||
|
||||
<span class="kn">from</span> <span class="nn">__future__</span> <span class="k">import</span> <span class="n">division</span>
|
||||
<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="sa">r</span><span class="sd">"""</span>
|
||||
<span class="sd"> Integration of :math:`f(x)` using the trapezoidal rule</span>
|
||||
<span class="sd"> (Simpson's rule, Kepler's rule).</span>
|
||||
|
||||
<span class="sd"> de: Trapezregel, Simpsonregel (Thomas Simpson), Keplersche</span>
|
||||
<span class="sd"> Fassregel (Johannes Kepler)</span>
|
||||
|
||||
<span class="sd"> :param f: function to integrate.</span>
|
||||
<span class="sd"> :type f: function or list</span>
|
||||
<span class="sd"> :param a: lower limit of integration (default = 0).</span>
|
||||
<span class="sd"> :type a: float</span>
|
||||
<span class="sd"> :param b: upper limit of integration (default = 1).</span>
|
||||
<span class="sd"> :type b: float</span>
|
||||
<span class="sd"> :param N: specify the number of subintervals.</span>
|
||||
<span class="sd"> :type N: int</span>
|
||||
<span class="sd"> :param x: variable of integration, necessary if f is a list</span>
|
||||
<span class="sd"> (default = None).</span>
|
||||
<span class="sd"> :type x: list</span>
|
||||
<span class="sd"> :param verbose: print information (default = False)</span>
|
||||
<span class="sd"> :type verbose: bool</span>
|
||||
|
||||
<span class="sd"> :returns: the definite integral as approximated by trapezoidal</span>
|
||||
<span class="sd"> rule.</span>
|
||||
<span class="sd"> :rtype: float</span>
|
||||
|
||||
<span class="sd"> The trapezoidal rule approximates the integral by the area of a</span>
|
||||
<span class="sd"> trapezoid with base h=b-a and sides equal to the values of the</span>
|
||||
<span class="sd"> integrand at the two end points.</span>
|
||||
|
||||
<span class="sd"> .. math::</span>
|
||||
<span class="sd"> f_n(x) = f(a)+\frac{f(b)-f(a)}{b-a}(x-a)</span>
|
||||
|
||||
<span class="sd"> .. math::</span>
|
||||
<span class="sd"> I &= \int\limits_a^b f(x) \,\mathrm{d}x \\</span>
|
||||
<span class="sd"> I &\approx \int\limits_a^b f_n(x) \,\mathrm{d}x \\</span>
|
||||
<span class="sd"> &= \int\limits_a^b</span>
|
||||
<span class="sd"> \left( f(a)+\frac{f(b)-f(a)}{b-a}(x-a) \right)</span>
|
||||
<span class="sd"> \mathrm{d}x \\</span>
|
||||
<span class="sd"> &= \left.\left( f(a)-a\frac{f(b)-f(a)}{b-a} \right)</span>
|
||||
<span class="sd"> x \right\vert_a^b +</span>
|
||||
<span class="sd"> \left. \frac{f(b)-f(a)}{b-a} \frac{x^2}{2}</span>
|
||||
<span class="sd"> \right\vert_a^b \\</span>
|
||||
<span class="sd"> &= \frac{b-a}{2}\left[f(a)+f(b)\right]</span>
|
||||
|
||||
<span class="sd"> The composite trapezium rule. If the interval is divided into n</span>
|
||||
<span class="sd"> segments (not necessarily equal)</span>
|
||||
|
||||
<span class="sd"> .. math::</span>
|
||||
<span class="sd"> a = x_0 \leq x_1 \leq x_2 \leq \ldots \leq x_n = b</span>
|
||||
|
||||
<span class="sd"> .. math::</span>
|
||||
<span class="sd"> I &\approx \sum\limits_{i=0}^{n-1} \frac{1}{2} (x_{i+1}-x_i)</span>
|
||||
<span class="sd"> \left[f(x_{i+1})+f(x_i)\right] \\</span>
|
||||
|
||||
<span class="sd"> Special Case (Equaliy spaced base points)</span>
|
||||
|
||||
<span class="sd"> .. math::</span>
|
||||
<span class="sd"> x_{i+1}-x_i = h \quad \forall i</span>
|
||||
|
||||
<span class="sd"> .. math::</span>
|
||||
<span class="sd"> I \approx h \left\{ \frac{1}{2} \left[f(x_0)+f(x_n)\right] +</span>
|
||||
<span class="sd"> \sum\limits_{i=1}^{n-1} f(x_i) \right\}</span>
|
||||
|
||||
<span class="sd"> .. rubric:: Example</span>
|
||||
|
||||
<span class="sd"> .. math::</span>
|
||||
<span class="sd"> I &= \int\limits_a^b f(x) \,\mathrm{d}x \\</span>
|
||||
<span class="sd"> f(x) &= x^2 \\</span>
|
||||
<span class="sd"> a &= 0 \\</span>
|
||||
<span class="sd"> b &= 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"> >>> f = lambda(x): x**2</span>
|
||||
<span class="sd"> >>> trapez(f, 0, 1, 1)</span>
|
||||
<span class="sd"> 0.5</span>
|
||||
<span class="sd"> >>> trapez(f, 0, 1, 10)</span>
|
||||
<span class="sd"> 0.3350000000000001</span>
|
||||
<span class="sd"> >>> trapez(f, 0, 1, 100)</span>
|
||||
<span class="sd"> 0.33335000000000004</span>
|
||||
<span class="sd"> """</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">'__call__'</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="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="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>
|
||||
|
||||
<span class="k">return</span> <span class="n">T</span></div>
|
||||
|
||||
|
||||
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">'__main__'</span><span class="p">:</span>
|
||||
<span class="n">func</span> <span class="o">=</span> <span class="k">lambda</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="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="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="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>
|
||||
|
||||
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|
||||
|
||||
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|
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|
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523
docs/build/html/_modules/numerical/ode.html
vendored
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vendored
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<h1>Source code for numerical.ode</h1><div class="highlight"><pre>
|
||||
<span></span><span class="ch">#!/usr/bin/env python</span>
|
||||
<span class="c1"># -*- coding: utf-8 -*-</span>
|
||||
<span class="sd">"""Numerical solver of ordinary differential equations.</span>
|
||||
|
||||
<span class="sd">Solves the initial value problem for systems of first order ordinary differential</span>
|
||||
<span class="sd">equations.</span>
|
||||
|
||||
<span class="sd">:Date: 2015-09-21</span>
|
||||
|
||||
<span class="sd">.. module:: ode</span>
|
||||
<span class="sd"> :platform: *nix, Windows</span>
|
||||
<span class="sd"> :synopsis: Numerical solver.</span>
|
||||
|
||||
<span class="sd">.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de></span>
|
||||
<span class="sd">"""</span>
|
||||
|
||||
<span class="kn">from</span> <span class="nn">__future__</span> <span class="k">import</span> <span class="n">division</span><span class="p">,</span> <span class="n">print_function</span>
|
||||
<span class="kn">from</span> <span class="nn">numpy</span> <span class="k">import</span> <span class="n">array</span><span class="p">,</span> <span class="n">isnan</span><span class="p">,</span> <span class="nb">sum</span><span class="p">,</span> <span class="n">zeros</span><span class="p">,</span> <span class="n">dot</span>
|
||||
<span class="kn">from</span> <span class="nn">numpy.linalg</span> <span class="k">import</span> <span class="n">norm</span><span class="p">,</span> <span class="n">inv</span>
|
||||
|
||||
<div class="viewcode-block" id="e1"><a class="viewcode-back" href="../../numerical.html#numerical.ode.e1">[docs]</a><span class="k">def</span> <span class="nf">e1</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">*</span><span class="n">p</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="sa">r</span><span class="sd">"""Explicit first-order method /</span>
|
||||
<span class="sd"> (standard, or forward) Euler method /</span>
|
||||
<span class="sd"> Runge-Kutta 1st order method.</span>
|
||||
|
||||
<span class="sd"> de:</span>
|
||||
<span class="sd"> Euler'sche Polygonzugverfahren / explizite Euler-Verfahren /</span>
|
||||
<span class="sd"> Euler-Cauchy-Verfahren / Euler-vorwärts-Verfahren</span>
|
||||
|
||||
<span class="sd"> :param f: the function to solve</span>
|
||||
<span class="sd"> :type f: function</span>
|
||||
<span class="sd"> :param x0: initial condition</span>
|
||||
<span class="sd"> :type x0: list</span>
|
||||
<span class="sd"> :param t: time</span>
|
||||
<span class="sd"> :type t: list</span>
|
||||
<span class="sd"> :param `*p`: parameters of the function (thickness, diameter, ...)</span>
|
||||
<span class="sd"> :param verbose: print information (default = False)</span>
|
||||
<span class="sd"> :type verbose: bool</span>
|
||||
|
||||
<span class="sd"> Approximate the solution of the initial value problem</span>
|
||||
|
||||
<span class="sd"> .. math ::</span>
|
||||
<span class="sd"> \dot{x} &= f(t,x) \\</span>
|
||||
<span class="sd"> x(t_0) &= x_0</span>
|
||||
|
||||
<span class="sd"> Choose a value h for the size of every step and set</span>
|
||||
|
||||
<span class="sd"> .. math ::</span>
|
||||
<span class="sd"> t_i = t_0 + i h ~,\quad i=1,2,\ldots,n</span>
|
||||
|
||||
<span class="sd"> The derivative of the solution is approximated as the forward difference</span>
|
||||
<span class="sd"> equation</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> .. math ::</span>
|
||||
<span class="sd"> \dot{x}_i = f(t_i, x_i) = \frac{x_{i+1} - x_i}{t_{i+1}-t_i}</span>
|
||||
|
||||
<span class="sd"> Therefore one step :math:`h` of the Euler method from :math:`t_i` to</span>
|
||||
<span class="sd"> :math:`t_{i+1}` is</span>
|
||||
|
||||
<span class="sd"> .. math ::</span>
|
||||
<span class="sd"> x_{i+1} &= x_i + (t_{i+1}-t_i) f(t_i, x_i) \\</span>
|
||||
<span class="sd"> x_{i+1} &= x_i + h f(t_i, x_i) \\</span>
|
||||
|
||||
<span class="sd"> Example 1:</span>
|
||||
|
||||
<span class="sd"> .. math ::</span>
|
||||
<span class="sd"> m\ddot{u} + d\dot{u} + ku = f(t) \\</span>
|
||||
<span class="sd"> \ddot{u} = m^{-1}(f(t) - d\dot{u} - ku) \\</span>
|
||||
|
||||
<span class="sd"> with</span>
|
||||
|
||||
<span class="sd"> .. math ::</span>
|
||||
<span class="sd"> x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\</span>
|
||||
<span class="sd"> x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\</span>
|
||||
|
||||
<span class="sd"> becomes</span>
|
||||
|
||||
<span class="sd"> .. math ::</span>
|
||||
<span class="sd"> \dot{x}_1 &= x_2 \\</span>
|
||||
<span class="sd"> \dot{x}_2 &= m^{-1}(f(t) - d x_2 - k x_1) \\</span>
|
||||
|
||||
<span class="sd"> or</span>
|
||||
|
||||
<span class="sd"> .. math ::</span>
|
||||
<span class="sd"> \dot{x} &= f(t,x) \\</span>
|
||||
<span class="sd"> \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &=</span>
|
||||
<span class="sd"> \begin{bmatrix} x_2 \\ m^{-1}(f(t) - d x_2 - k x_1) \end{bmatrix} \\</span>
|
||||
<span class="sd"> &=</span>
|
||||
<span class="sd"> \begin{bmatrix} 0 \\ m^{-1} f(t) \end{bmatrix} +</span>
|
||||
<span class="sd"> \begin{bmatrix} 0 & 1 \\ -m^{-1} k & -m^{-1} d \end{bmatrix}</span>
|
||||
<span class="sd"> \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}</span>
|
||||
|
||||
<span class="sd"> Example 2:</span>
|
||||
|
||||
<span class="sd"> .. math ::</span>
|
||||
<span class="sd"> m(u)\ddot{u} + d(u,\dot{u})\dot{u} + k(u)u = f(t) \\</span>
|
||||
<span class="sd"> \ddot{u} = m^{-1}(u)(f(t) - d(u,\dot{u})\dot{u} - k(u)u) \\</span>
|
||||
|
||||
<span class="sd"> with</span>
|
||||
|
||||
<span class="sd"> .. math ::</span>
|
||||
<span class="sd"> x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\</span>
|
||||
<span class="sd"> x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\</span>
|
||||
|
||||
<span class="sd"> becomes</span>
|
||||
|
||||
<span class="sd"> .. math ::</span>
|
||||
<span class="sd"> \dot{x}_1 &= x_2 \\</span>
|
||||
<span class="sd"> \dot{x}_2 &= m^{-1}(x_1)(f(t) - d(x_1,x_2) x_2 - k(x_1) x_1) \\</span>
|
||||
|
||||
<span class="sd"> or</span>
|
||||
|
||||
<span class="sd"> .. math ::</span>
|
||||
<span class="sd"> \dot{x} &= f(t,x) \\</span>
|
||||
<span class="sd"> \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &=</span>
|
||||
<span class="sd"> \begin{bmatrix} x_2 \\ m^{-1}(x_1)(f(t) - d(x_1,x_2) x_2 - k(x_1) x_1) \end{bmatrix} \\</span>
|
||||
<span class="sd"> &=</span>
|
||||
<span class="sd"> \begin{bmatrix} 0 \\ m^{-1}(x_1) f(t) \end{bmatrix} +</span>
|
||||
<span class="sd"> \begin{bmatrix} 0 & 1 \\ -m^{-1}(x_1) k(x_1) & -m^{-1} d(x_1,x_2) \end{bmatrix}</span>
|
||||
<span class="sd"> \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}</span>
|
||||
|
||||
<span class="sd"> The Euler method is a first-order method,</span>
|
||||
<span class="sd"> which means that the local error (error per step) is proportional to the</span>
|
||||
<span class="sd"> square of the step size, and the global error (error at a given time) is</span>
|
||||
<span class="sd"> proportional to the step size.</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)))</span> <span class="c1"># Preallocate array</span>
|
||||
<span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">x0</span> <span class="c1"># Initial condition gives solution at first t</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span> <span class="c1"># Calculation loop</span>
|
||||
<span class="n">Dt</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
|
||||
<span class="n">dxdt</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,:],</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="o">*</span><span class="n">p</span><span class="p">))</span>
|
||||
<span class="n">x</span><span class="p">[</span><span class="n">i</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">i</span><span class="p">,:]</span> <span class="o">+</span> <span class="n">dxdt</span><span class="o">*</span><span class="n">Dt</span> <span class="c1"># Approximate solution at next value of x</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="s1">'Numerical integration of ODE using explicit first-order method (Euler / Runge-Kutta) was successful.'</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">x</span></div>
|
||||
|
||||
<div class="viewcode-block" id="e2"><a class="viewcode-back" href="../../numerical.html#numerical.ode.e2">[docs]</a><span class="k">def</span> <span class="nf">e2</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">*</span><span class="n">p</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="sa">r</span><span class="sd">"""Explicit second-order method / Runge-Kutta 2nd order method.</span>
|
||||
|
||||
<span class="sd"> :param f: the function to solve</span>
|
||||
<span class="sd"> :type f: function</span>
|
||||
<span class="sd"> :param x0: initial condition</span>
|
||||
<span class="sd"> :type x0: list</span>
|
||||
<span class="sd"> :param t: time</span>
|
||||
<span class="sd"> :type t: list</span>
|
||||
<span class="sd"> :param `*p`: parameters of the function (thickness, diameter, ...)</span>
|
||||
<span class="sd"> :param verbose: print information (default = False)</span>
|
||||
<span class="sd"> :type verbose: bool</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)))</span> <span class="c1"># Preallocate array</span>
|
||||
<span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">x0</span> <span class="c1"># Initial condition gives solution at first t</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span> <span class="c1"># Calculation loop</span>
|
||||
<span class="n">Dt</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
|
||||
<span class="n">k_1</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,:],</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="o">*</span><span class="n">p</span><span class="p">))</span>
|
||||
<span class="n">k_2</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span><span class="o">+</span><span class="mf">0.5</span><span class="o">*</span><span class="n">Dt</span><span class="o">*</span><span class="n">k_1</span><span class="p">,</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">+</span><span class="mf">0.5</span><span class="o">*</span><span class="n">Dt</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">))</span>
|
||||
<span class="n">x</span><span class="p">[</span><span class="n">i</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">i</span><span class="p">,:]</span> <span class="o">+</span> <span class="n">k_2</span><span class="o">*</span><span class="n">Dt</span> <span class="c1"># Approximate solution at next value of x</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="s1">'Numerical integration of ODE using explicit 2th-order method (Runge-Kutta) was successful.'</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">x</span></div>
|
||||
|
||||
<div class="viewcode-block" id="e4"><a class="viewcode-back" href="../../numerical.html#numerical.ode.e4">[docs]</a><span class="k">def</span> <span class="nf">e4</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">*</span><span class="n">p</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="sa">r</span><span class="sd">"""Explicit fourth-order method / Runge-Kutta 4th order method.</span>
|
||||
|
||||
<span class="sd"> :param f: the function to solve</span>
|
||||
<span class="sd"> :type f: function</span>
|
||||
<span class="sd"> :param x0: initial condition</span>
|
||||
<span class="sd"> :type x0: list</span>
|
||||
<span class="sd"> :param t: time</span>
|
||||
<span class="sd"> :type t: list</span>
|
||||
<span class="sd"> :param `*p`: parameters of the function (thickness, diameter, ...)</span>
|
||||
<span class="sd"> :param verbose: print information (default = False)</span>
|
||||
<span class="sd"> :type verbose: bool</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)))</span> <span class="c1"># Preallocate array</span>
|
||||
<span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">x0</span> <span class="c1"># Initial condition</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span> <span class="c1"># Calculation loop</span>
|
||||
<span class="n">Dt</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
|
||||
<span class="n">k_1</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,:],</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="o">*</span><span class="n">p</span><span class="p">))</span>
|
||||
<span class="n">k_2</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span><span class="o">+</span><span class="mf">0.5</span><span class="o">*</span><span class="n">Dt</span><span class="o">*</span><span class="n">k_1</span><span class="p">,</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">+</span><span class="mf">0.5</span><span class="o">*</span><span class="n">Dt</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">))</span>
|
||||
<span class="n">k_3</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span><span class="o">+</span><span class="mf">0.5</span><span class="o">*</span><span class="n">Dt</span><span class="o">*</span><span class="n">k_2</span><span class="p">,</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">+</span><span class="mf">0.5</span><span class="o">*</span><span class="n">Dt</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">))</span>
|
||||
<span class="n">k_4</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span><span class="o">+</span><span class="n">k_3</span><span class="o">*</span><span class="n">Dt</span><span class="p">,</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">+</span><span class="n">Dt</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">))</span>
|
||||
<span class="n">x</span><span class="p">[</span><span class="n">i</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">i</span><span class="p">,:]</span> <span class="o">+</span> <span class="mf">1.</span><span class="o">/</span><span class="mi">6</span><span class="o">*</span><span class="p">(</span><span class="n">k_1</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">k_2</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">k_3</span><span class="o">+</span><span class="n">k_4</span><span class="p">)</span><span class="o">*</span><span class="n">Dt</span> <span class="c1"># Approximate solution at next value of x</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="s1">'Numerical integration of ODE using explicit 4th-order method (Runge-Kutta) was successful.'</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">x</span></div>
|
||||
|
||||
<div class="viewcode-block" id="dxdt_Dt"><a class="viewcode-back" href="../../numerical.html#numerical.ode.dxdt_Dt">[docs]</a><span class="k">def</span> <span class="nf">dxdt_Dt</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">Dt</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">):</span>
|
||||
<span class="sa">r</span><span class="sd">"""</span>
|
||||
<span class="sd"> :param f: :math:`f = \dot{x}`</span>
|
||||
<span class="sd"> :type f: function</span>
|
||||
<span class="sd"> :param Dt: :math:`\Delta{t}`</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> :returns: :math:`\Delta x = \dot{x} \Delta t`</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="k">return</span> <span class="n">array</span><span class="p">(</span><span class="n">dxdt</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">))</span> <span class="o">*</span> <span class="n">Dt</span></div>
|
||||
|
||||
<div class="viewcode-block" id="fixed_point_iteration"><a class="viewcode-back" href="../../numerical.html#numerical.ode.fixed_point_iteration">[docs]</a><span class="k">def</span> <span class="nf">fixed_point_iteration</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">xi</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">max_iterations</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-9</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="sa">r</span><span class="sd">"""</span>
|
||||
<span class="sd"> :param f: the function to iterate :math:`f = \Delta{x}(t)`</span>
|
||||
<span class="sd"> :type f: function</span>
|
||||
<span class="sd"> :param xi: initial condition :math:`x_i`</span>
|
||||
<span class="sd"> :type xi: list</span>
|
||||
<span class="sd"> :param t: time :math:`t`</span>
|
||||
<span class="sd"> :type t: float</span>
|
||||
<span class="sd"> :param `*p`: parameters of the function (thickness, diameter, ...)</span>
|
||||
<span class="sd"> :param max_iterations: maximum number of iterations</span>
|
||||
<span class="sd"> :type max_iterations: int</span>
|
||||
<span class="sd"> :param tol: tolerance against residuum (default = 1e-9)</span>
|
||||
<span class="sd"> :type tol: float</span>
|
||||
<span class="sd"> :param verbose: print information (default = False)</span>
|
||||
<span class="sd"> :type verbose: bool</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> :returns: :math:`x_{i+1}`</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> .. math ::</span>
|
||||
<span class="sd"> x_{i+1} = x_i + \Delta x</span>
|
||||
|
||||
<span class="sd"> .. seealso::</span>
|
||||
<span class="sd"> :meth:`dxdt_Dt` for :math:`\Delta x`</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">xi</span> <span class="o">=</span> <span class="n">x0</span>
|
||||
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">max_iterations</span><span class="p">):</span> <span class="c1"># Fixed-point iteration</span>
|
||||
<span class="n">Dx</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">xi</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">))</span>
|
||||
<span class="n">xi1</span> <span class="o">=</span> <span class="n">x0</span> <span class="o">+</span> <span class="n">Dx</span> <span class="c1"># Approximate solution at next value of x</span>
|
||||
<span class="n">residuum</span> <span class="o">=</span> <span class="n">norm</span><span class="p">(</span><span class="n">xi1</span><span class="o">-</span><span class="n">xi</span><span class="p">)</span><span class="o">/</span><span class="n">norm</span><span class="p">(</span><span class="n">xi1</span><span class="p">)</span>
|
||||
<span class="n">xi</span> <span class="o">=</span> <span class="n">xi1</span>
|
||||
<span class="k">if</span> <span class="n">residuum</span> <span class="o"><</span> <span class="n">tol</span><span class="p">:</span>
|
||||
<span class="k">break</span>
|
||||
<span class="n">iterations</span> <span class="o">=</span> <span class="n">j</span><span class="o">+</span><span class="mi">1</span> <span class="c1"># number beginning with 1 therefore + 1</span>
|
||||
<span class="k">return</span> <span class="n">xi</span><span class="p">,</span> <span class="n">iterations</span></div>
|
||||
|
||||
<div class="viewcode-block" id="i1n"><a class="viewcode-back" href="../../numerical.html#numerical.ode.i1n">[docs]</a><span class="k">def</span> <span class="nf">i1n</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">,</span> <span class="n">max_iterations</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-9</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">iterations</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="mi">1</span><span class="p">))</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)))</span> <span class="c1"># Preallocate array</span>
|
||||
<span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">x0</span> <span class="c1"># Initial condition gives solution at first t</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
|
||||
<span class="n">Dt</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
|
||||
<span class="n">xi</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
|
||||
<span class="n">Dx</span> <span class="o">=</span> <span class="n">dxdt_Dt</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">xi</span><span class="p">,</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">],</span> <span class="n">Dt</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">)</span>
|
||||
<span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,:],</span> <span class="n">iterations</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">fixed_point_iteration</span><span class="p">(</span><span class="n">Dx</span><span class="p">,</span> <span class="n">xi</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">max_iterations</span><span class="p">,</span> <span class="n">tol</span><span class="p">,</span> <span class="n">verbose</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="s1">'Numerical integration of ODE using implicite first-order method (Euler) was successful.'</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">iterations</span></div>
|
||||
|
||||
<div class="viewcode-block" id="i1"><a class="viewcode-back" href="../../numerical.html#numerical.ode.i1">[docs]</a><span class="k">def</span> <span class="nf">i1</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">,</span> <span class="n">max_iterations</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-9</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="sa">r</span><span class="sd">"""Implicite first-order method / backward Euler method.</span>
|
||||
|
||||
<span class="sd"> :param f: the function to solve</span>
|
||||
<span class="sd"> :type f: function</span>
|
||||
<span class="sd"> :param x0: initial condition</span>
|
||||
<span class="sd"> :type x0: list</span>
|
||||
<span class="sd"> :param t: time</span>
|
||||
<span class="sd"> :type t: list</span>
|
||||
<span class="sd"> :param `*p`: parameters of the function (thickness, diameter, ...)</span>
|
||||
<span class="sd"> :param max_iterations: maximum number of iterations</span>
|
||||
<span class="sd"> :type max_iterations: int</span>
|
||||
<span class="sd"> :param tol: tolerance against residuum (default = 1e-9)</span>
|
||||
<span class="sd"> :type tol: float</span>
|
||||
<span class="sd"> :param verbose: print information (default = False)</span>
|
||||
<span class="sd"> :type verbose: bool</span>
|
||||
<span class="sd"> </span>
|
||||
<span class="sd"> The backward Euler method has order one and is A-stable.</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">iterations</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="mi">1</span><span class="p">))</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)))</span> <span class="c1"># Preallocate array</span>
|
||||
<span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">x0</span> <span class="c1"># Initial condition gives solution at first t</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
|
||||
<span class="n">Dt</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
|
||||
<span class="n">xi</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
|
||||
<span class="c1"># x(i+1) = x(i) + f(x(i+1), t(i+1)), exact value of f(x(i+1), t(i+1)) is not</span>
|
||||
<span class="c1"># available therefor using Newton-Raphson method</span>
|
||||
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">max_iterations</span><span class="p">):</span> <span class="c1"># Fixed-point iteration</span>
|
||||
<span class="n">dxdt</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">xi</span><span class="p">,</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">],</span> <span class="o">*</span><span class="n">p</span><span class="p">))</span>
|
||||
<span class="n">xi1</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">+</span> <span class="n">dxdt</span><span class="o">*</span><span class="n">Dt</span> <span class="c1"># Approximate solution at next value of x</span>
|
||||
<span class="n">residuum</span> <span class="o">=</span> <span class="n">norm</span><span class="p">(</span><span class="n">xi1</span><span class="o">-</span><span class="n">xi</span><span class="p">)</span><span class="o">/</span><span class="n">norm</span><span class="p">(</span><span class="n">xi1</span><span class="p">)</span>
|
||||
<span class="n">xi</span> <span class="o">=</span> <span class="n">xi1</span>
|
||||
<span class="k">if</span> <span class="n">residuum</span> <span class="o"><</span> <span class="n">tol</span><span class="p">:</span>
|
||||
<span class="k">break</span>
|
||||
<span class="n">iterations</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">j</span><span class="o">+</span><span class="mi">1</span>
|
||||
<span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">xi</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="s1">'Numerical integration of ODE using implicite first-order method (Euler) was successful.'</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">iterations</span></div>
|
||||
|
||||
<div class="viewcode-block" id="newmark_newtonraphson"><a class="viewcode-back" href="../../numerical.html#numerical.ode.newmark_newtonraphson">[docs]</a><span class="k">def</span> <span class="nf">newmark_newtonraphson</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">xp0</span><span class="p">,</span> <span class="n">xpp0</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=.</span><span class="mi">5</span><span class="p">,</span> <span class="n">beta</span><span class="o">=.</span><span class="mi">25</span><span class="p">,</span> <span class="n">max_iterations</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-9</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="sa">r</span><span class="sd">"""Newmark method.</span>
|
||||
|
||||
<span class="sd"> :param f: the function to solve</span>
|
||||
<span class="sd"> :type f: function</span>
|
||||
<span class="sd"> :param x0: initial condition</span>
|
||||
<span class="sd"> :type x0: list</span>
|
||||
<span class="sd"> :param xp0: initial condition</span>
|
||||
<span class="sd"> :type xp0: list</span>
|
||||
<span class="sd"> :param xpp0: initial condition</span>
|
||||
<span class="sd"> :type xpp0: list</span>
|
||||
<span class="sd"> :param t: time</span>
|
||||
<span class="sd"> :type t: list</span>
|
||||
<span class="sd"> :param `*p`: parameters of the function (thickness, diameter, ...)</span>
|
||||
<span class="sd"> :param gamma: newmark parameter for velocity (default = 0.5)</span>
|
||||
<span class="sd"> :type gamma: float</span>
|
||||
<span class="sd"> :param beta: newmark parameter for displacement (default = 0.25)</span>
|
||||
<span class="sd"> :type beta: float</span>
|
||||
<span class="sd"> :param max_iterations: maximum number of iterations</span>
|
||||
<span class="sd"> :type max_iterations: int</span>
|
||||
<span class="sd"> :param tol: tolerance against residuum (default = 1e-9)</span>
|
||||
<span class="sd"> :type tol: float</span>
|
||||
<span class="sd"> :param verbose: print information (default = False)</span>
|
||||
<span class="sd"> :type verbose: bool</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">iterations</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="mi">1</span><span class="p">))</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)))</span> <span class="c1"># Preallocate array</span>
|
||||
<span class="n">xp</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">xp0</span><span class="p">)))</span> <span class="c1"># Preallocate array</span>
|
||||
<span class="n">xpp</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">xpp0</span><span class="p">)))</span> <span class="c1"># Preallocate array</span>
|
||||
<span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">x0</span> <span class="c1"># Initial condition gives solution at first t</span>
|
||||
<span class="n">xp</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">xp0</span> <span class="c1"># Initial condition gives solution at first t</span>
|
||||
<span class="n">xpp</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">xpp0</span> <span class="c1"># Initial condition gives solution at first t</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
|
||||
<span class="n">Dt</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
|
||||
|
||||
<span class="n">xi</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">xpi</span> <span class="o">=</span> <span class="n">xp</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">xppi</span> <span class="o">=</span> <span class="n">xpp</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">x1</span> <span class="o">=</span> <span class="n">xi</span>
|
||||
<span class="n">xp1</span> <span class="o">=</span> <span class="n">xpi</span>
|
||||
<span class="n">xpp1</span> <span class="o">=</span> <span class="n">xppi</span>
|
||||
<span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
|
||||
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">max_iterations</span><span class="p">):</span> <span class="c1"># Fixed-point iteration</span>
|
||||
<span class="c1">#dxdt = array(f(t[i+1], x1, p))</span>
|
||||
<span class="c1">#x11 = x[i,:] + dxdt*Dt # Approximate solution at next value of x</span>
|
||||
|
||||
<span class="n">N</span><span class="p">,</span> <span class="n">dN</span><span class="p">,</span> <span class="n">dNp</span><span class="p">,</span> <span class="n">dNpp</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x1</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,)</span><span class="o">.</span><span class="n">tolist</span><span class="p">(),</span>
|
||||
<span class="n">xp1</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,)</span><span class="o">.</span><span class="n">tolist</span><span class="p">(),</span> <span class="n">xpp1</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,)</span><span class="o">.</span><span class="n">tolist</span><span class="p">(),</span>
|
||||
<span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="o">*</span><span class="n">p</span><span class="p">)</span>
|
||||
<span class="k">if</span> <span class="n">isnan</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">dN</span><span class="p">))</span> <span class="ow">or</span> <span class="n">isnan</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">dNp</span><span class="p">))</span> <span class="ow">or</span> <span class="n">isnan</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">dNpp</span><span class="p">)):</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">'divergiert'</span><span class="p">)</span>
|
||||
<span class="k">break</span>
|
||||
|
||||
<span class="n">xpp11</span> <span class="o">=</span> <span class="n">xpp1</span> <span class="o">-</span> <span class="n">dot</span><span class="p">(</span><span class="n">inv</span><span class="p">(</span><span class="n">dNpp</span><span class="p">),</span> <span class="p">(</span><span class="n">N</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">dN</span><span class="p">,</span> <span class="p">(</span><span class="n">x1</span><span class="o">-</span><span class="n">xi</span><span class="p">))</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">dNp</span><span class="p">,</span> <span class="p">(</span><span class="n">xp1</span><span class="o">-</span><span class="n">xpi</span><span class="p">))))</span>
|
||||
<span class="n">xp1</span> <span class="o">=</span> <span class="n">xpi</span> <span class="o">+</span> <span class="n">Dt</span><span class="o">*</span><span class="p">(</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">gamma</span><span class="p">)</span><span class="o">*</span><span class="n">xppi</span> <span class="o">+</span> <span class="n">gamma</span><span class="o">*</span><span class="n">xpp11</span> <span class="p">)</span>
|
||||
<span class="n">x1</span> <span class="o">=</span> <span class="n">xi</span> <span class="o">+</span> <span class="n">Dt</span><span class="o">*</span><span class="n">xpi</span> <span class="o">+</span> <span class="n">Dt</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span> <span class="p">(</span><span class="o">.</span><span class="mi">5</span><span class="o">-</span><span class="n">beta</span><span class="p">)</span><span class="o">*</span><span class="n">xppi</span> <span class="o">+</span> <span class="n">beta</span><span class="o">*</span><span class="n">xpp11</span> <span class="p">)</span>
|
||||
|
||||
<span class="n">residuum</span> <span class="o">=</span> <span class="n">norm</span><span class="p">(</span><span class="n">xpp11</span><span class="o">-</span><span class="n">xpp1</span><span class="p">)</span><span class="o">/</span><span class="n">norm</span><span class="p">(</span><span class="n">xpp11</span><span class="p">)</span>
|
||||
<span class="n">xpp1</span> <span class="o">=</span> <span class="n">xpp11</span>
|
||||
<span class="k">if</span> <span class="n">residuum</span> <span class="o"><</span> <span class="n">tol</span><span class="p">:</span>
|
||||
<span class="k">break</span>
|
||||
<span class="n">iterations</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">j</span><span class="o">+</span><span class="mi">1</span>
|
||||
|
||||
<span class="n">xpp</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">xpp1</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,)</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
|
||||
<span class="n">xp</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">xp1</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,)</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
|
||||
<span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">x1</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,)</span><span class="o">.</span><span class="n">tolist</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="s1">'Numerical integration of ODE using explicite newmark method was successful.'</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">xp</span><span class="p">,</span> <span class="n">xpp</span><span class="p">,</span> <span class="n">iterations</span></div>
|
||||
<span class="c1"># x = concatenate((x, xp, xpp), axis=1)</span>
|
||||
|
||||
<div class="viewcode-block" id="newmark_newtonraphson_rdk"><a class="viewcode-back" href="../../numerical.html#numerical.ode.newmark_newtonraphson_rdk">[docs]</a><span class="k">def</span> <span class="nf">newmark_newtonraphson_rdk</span><span class="p">(</span><span class="n">fnm</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">xp0</span><span class="p">,</span> <span class="n">xpp0</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=.</span><span class="mi">5</span><span class="p">,</span> <span class="n">beta</span><span class="o">=.</span><span class="mi">25</span><span class="p">,</span> <span class="n">maxIterations</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-9</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="sa">r</span><span class="sd">"""Newmark method.</span>
|
||||
|
||||
<span class="sd"> :param f: the function to solve</span>
|
||||
<span class="sd"> :type f: function</span>
|
||||
<span class="sd"> :param x0: initial condition</span>
|
||||
<span class="sd"> :type x0: list</span>
|
||||
<span class="sd"> :param xp0: initial condition</span>
|
||||
<span class="sd"> :type xp0: list</span>
|
||||
<span class="sd"> :param xpp0: initial condition</span>
|
||||
<span class="sd"> :type xpp0: list</span>
|
||||
<span class="sd"> :param t: time</span>
|
||||
<span class="sd"> :type t: list</span>
|
||||
<span class="sd"> :param `*p`: parameters of the function (thickness, diameter, ...)</span>
|
||||
<span class="sd"> :param gamma: newmark parameter for velocity (default = 0.5)</span>
|
||||
<span class="sd"> :type gamma: float</span>
|
||||
<span class="sd"> :param beta: newmark parameter for displacement (default = 0.25)</span>
|
||||
<span class="sd"> :type beta: float</span>
|
||||
<span class="sd"> :param max_iterations: maximum number of iterations</span>
|
||||
<span class="sd"> :type max_iterations: int</span>
|
||||
<span class="sd"> :param tol: tolerance against residuum (default = 1e-9)</span>
|
||||
<span class="sd"> :type tol: float</span>
|
||||
<span class="sd"> :param verbose: print information (default = False)</span>
|
||||
<span class="sd"> :type verbose: bool</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">iterations</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="mi">1</span><span class="p">))</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)))</span> <span class="c1"># Preallocate array</span>
|
||||
<span class="n">xp</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">xp0</span><span class="p">)))</span> <span class="c1"># Preallocate array</span>
|
||||
<span class="n">xpp</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">xpp0</span><span class="p">)))</span> <span class="c1"># Preallocate array</span>
|
||||
<span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">x0</span> <span class="c1"># Initial condition gives solution at first t</span>
|
||||
<span class="n">xp</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">xp0</span> <span class="c1"># Initial condition gives solution at first t</span>
|
||||
<span class="n">xpp</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">xpp0</span> <span class="c1"># Initial condition gives solution at first t</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
|
||||
<span class="n">Dt</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
|
||||
|
||||
<span class="n">rm</span><span class="p">,</span> <span class="n">rmx</span><span class="p">,</span> <span class="n">rmxpp</span><span class="p">,</span> <span class="n">rd</span><span class="p">,</span> <span class="n">rdx</span><span class="p">,</span> <span class="n">rdxp</span><span class="p">,</span> <span class="n">rk</span><span class="p">,</span> <span class="n">rkx</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">fnm</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,:],</span> <span class="n">xp</span><span class="p">[</span><span class="n">i</span><span class="p">,:],</span> <span class="n">xpp</span><span class="p">[</span><span class="n">i</span><span class="p">,:],</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="o">*</span><span class="n">p</span><span class="p">)</span>
|
||||
|
||||
<span class="n">xi</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">xpi</span> <span class="o">=</span> <span class="n">xp</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">xppi</span> <span class="o">=</span> <span class="n">xpp</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">x1</span> <span class="o">=</span> <span class="n">xi</span>
|
||||
<span class="n">xp1</span> <span class="o">=</span> <span class="n">xpi</span>
|
||||
<span class="n">xpp1</span> <span class="o">=</span> <span class="n">xppi</span>
|
||||
<span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
|
||||
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">maxIterations</span><span class="p">):</span> <span class="c1"># Fixed-point iteration</span>
|
||||
<span class="c1">#dxdt = array(f(t[i+1], x1, p))</span>
|
||||
<span class="c1">#x11 = x[i,:] + dxdt*Dt # Approximate solution at next value of x</span>
|
||||
|
||||
<span class="n">r</span> <span class="o">=</span> <span class="p">(</span><span class="n">rmx</span><span class="o">+</span><span class="n">rdx</span><span class="o">+</span><span class="n">rkx</span><span class="p">)</span><span class="o">*</span><span class="n">Dt</span><span class="o">**</span><span class="mf">2.</span><span class="o">/</span><span class="mi">4</span> <span class="o">+</span> <span class="n">rdxp</span><span class="o">*</span><span class="n">Dt</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">rmxpp</span>
|
||||
<span class="n">rp</span> <span class="o">=</span> <span class="n">f</span> <span class="o">-</span> <span class="p">(</span><span class="n">rm</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">rmx</span><span class="p">,</span> <span class="p">(</span><span class="n">Dt</span><span class="o">*</span><span class="n">xpi</span><span class="o">+</span><span class="n">Dt</span><span class="o">**</span><span class="mf">2.</span><span class="o">/</span><span class="mi">4</span><span class="o">*</span><span class="n">xppi</span><span class="p">))</span> <span class="o">-</span> <span class="n">dot</span><span class="p">(</span><span class="n">rmxpp</span><span class="p">,</span> <span class="n">xppi</span><span class="p">)</span> <span class="o">+</span> \
|
||||
<span class="n">rd</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">rdx</span><span class="p">,</span> <span class="p">(</span><span class="n">Dt</span><span class="o">*</span><span class="n">xpi</span><span class="o">+</span><span class="n">Dt</span><span class="o">**</span><span class="mf">2.</span><span class="o">/</span><span class="mi">4</span><span class="o">*</span><span class="n">xppi</span><span class="p">))</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">rdxp</span><span class="p">,</span> <span class="n">Dt</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="n">xppi</span><span class="p">)</span> <span class="o">+</span> \
|
||||
<span class="n">rk</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">rkx</span><span class="p">,</span> <span class="p">(</span><span class="n">Dt</span><span class="o">*</span><span class="n">xpi</span><span class="o">+</span><span class="n">Dt</span><span class="o">**</span><span class="mf">2.</span><span class="o">/</span><span class="mi">4</span><span class="o">*</span><span class="n">xppi</span><span class="p">))</span> <span class="p">)</span>
|
||||
<span class="n">xpp11</span> <span class="o">=</span> <span class="n">dot</span><span class="p">(</span><span class="n">inv</span><span class="p">(</span><span class="n">r</span><span class="p">),</span> <span class="n">rp</span><span class="p">)</span>
|
||||
<span class="n">xp1</span> <span class="o">=</span> <span class="n">xpi</span> <span class="o">+</span> <span class="n">Dt</span><span class="o">*</span><span class="p">(</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">gamma</span><span class="p">)</span><span class="o">*</span><span class="n">xppi</span> <span class="o">+</span> <span class="n">gamma</span><span class="o">*</span><span class="n">xpp11</span> <span class="p">)</span>
|
||||
<span class="n">x1</span> <span class="o">=</span> <span class="n">xi</span> <span class="o">+</span> <span class="n">Dt</span><span class="o">*</span><span class="n">xpi</span> <span class="o">+</span> <span class="n">Dt</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span> <span class="p">(</span><span class="o">.</span><span class="mi">5</span><span class="o">-</span><span class="n">beta</span><span class="p">)</span><span class="o">*</span><span class="n">xppi</span> <span class="o">+</span> <span class="n">beta</span><span class="o">*</span><span class="n">xpp11</span> <span class="p">)</span>
|
||||
|
||||
<span class="n">residuum</span> <span class="o">=</span> <span class="n">norm</span><span class="p">(</span><span class="n">xpp11</span><span class="o">-</span><span class="n">xpp1</span><span class="p">)</span><span class="o">/</span><span class="n">norm</span><span class="p">(</span><span class="n">xpp11</span><span class="p">)</span>
|
||||
<span class="n">xpp1</span> <span class="o">=</span> <span class="n">xpp11</span>
|
||||
<span class="k">if</span> <span class="n">residuum</span> <span class="o"><</span> <span class="n">tol</span><span class="p">:</span>
|
||||
<span class="k">break</span>
|
||||
<span class="n">iterations</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">j</span><span class="o">+</span><span class="mi">1</span>
|
||||
|
||||
<span class="n">xpp</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">xpp1</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,)</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
|
||||
<span class="n">xp</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">xp1</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,)</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
|
||||
<span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">x1</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,)</span><span class="o">.</span><span class="n">tolist</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="s1">'Numerical integration of ODE using explicite newmark method was successful.'</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">xp</span><span class="p">,</span> <span class="n">xpp</span><span class="p">,</span> <span class="n">iterations</span></div>
|
||||
<span class="c1"># x = concatenate((x, xp, xpp), axis=1)</span>
|
||||
</pre></div>
|
||||
|
||||
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|
||||
|
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219
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vendored
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<h1>Source code for numerical.ode_model</h1><div class="highlight"><pre>
|
||||
<span></span><span class="ch">#!/usr/bin/env python</span>
|
||||
<span class="c1"># -*- coding: utf-8 -*-</span>
|
||||
<span class="sd">"""Mathmatical models governed by ordinary differential equations.</span>
|
||||
|
||||
<span class="sd">Describes initial value problems as systems of first order ordinary differential</span>
|
||||
<span class="sd">equations.</span>
|
||||
|
||||
<span class="sd">:Date: 2019-05-25</span>
|
||||
|
||||
<span class="sd">.. module:: ode_model</span>
|
||||
<span class="sd"> :platform: *nix, Windows</span>
|
||||
<span class="sd"> :synopsis: Models of ordinary differential equations.</span>
|
||||
|
||||
<span class="sd">.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de></span>
|
||||
<span class="sd">"""</span>
|
||||
<span class="kn">from</span> <span class="nn">__future__</span> <span class="k">import</span> <span class="n">division</span><span class="p">,</span> <span class="n">print_function</span>
|
||||
<span class="kn">from</span> <span class="nn">numpy</span> <span class="k">import</span> <span class="n">array</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">dot</span><span class="p">,</span> <span class="n">square</span>
|
||||
<span class="kn">from</span> <span class="nn">numpy.linalg</span> <span class="k">import</span> <span class="n">inv</span>
|
||||
|
||||
<div class="viewcode-block" id="disk"><a class="viewcode-back" href="../../numerical.html#numerical.ode_model.disk">[docs]</a><span class="k">def</span> <span class="nf">disk</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">):</span>
|
||||
<span class="sd">"""Rotation of an eccentric disk.</span>
|
||||
|
||||
<span class="sd"> :param x: values of the function</span>
|
||||
<span class="sd"> :type x: list</span>
|
||||
<span class="sd"> :param t: time</span>
|
||||
<span class="sd"> :type t: list</span>
|
||||
<span class="sd"> :param `*p`: parameters of the function</span>
|
||||
|
||||
<span class="sd"> * diameter</span>
|
||||
<span class="sd"> * eccentricity</span>
|
||||
<span class="sd"> * torque</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">qp1</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
|
||||
<span class="n">qp2</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span>
|
||||
<span class="n">qp3</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span>
|
||||
<span class="n">M</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</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="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span> <span class="p">[</span><span class="mi">0</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="n">y</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span><span class="o">+</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> \
|
||||
<span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span><span class="o">+</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> \
|
||||
<span class="p">[</span><span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">p</span><span class="p">[</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="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">+</span><span class="n">p</span><span class="p">[</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="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])]])</span>
|
||||
<span class="n">qp46</span> <span class="o">=</span> <span class="n">dot</span><span class="p">(</span><span class="n">inv</span><span class="p">(</span><span class="n">M</span><span class="p">),</span> <span class="n">y</span><span class="p">)</span>
|
||||
<span class="n">qp4</span><span class="p">,</span> <span class="n">qp5</span><span class="p">,</span> <span class="n">qp6</span> <span class="o">=</span> <span class="n">qp46</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,)</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span> <span class="c1"># 2d array to 1d array to list</span>
|
||||
<span class="k">return</span> <span class="n">qp1</span><span class="p">,</span> <span class="n">qp2</span><span class="p">,</span> <span class="n">qp3</span><span class="p">,</span> <span class="n">qp4</span><span class="p">,</span> <span class="n">qp5</span><span class="p">,</span> <span class="n">qp6</span></div>
|
||||
|
||||
<div class="viewcode-block" id="disk_nm"><a class="viewcode-back" href="../../numerical.html#numerical.ode_model.disk_nm">[docs]</a><span class="k">def</span> <span class="nf">disk_nm</span><span class="p">(</span><span class="n">xn</span><span class="p">,</span> <span class="n">xpn</span><span class="p">,</span> <span class="n">xppn</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">):</span>
|
||||
<span class="sd">"""Rotation of an eccentric disk.</span>
|
||||
|
||||
<span class="sd"> :param xn: values of the function</span>
|
||||
<span class="sd"> :type xn: list</span>
|
||||
<span class="sd"> :param xpn: first derivative values of the function</span>
|
||||
<span class="sd"> :type xpn: list</span>
|
||||
<span class="sd"> :param xppn: second derivative values of the function</span>
|
||||
<span class="sd"> :type xppn: list</span>
|
||||
<span class="sd"> :param t: time</span>
|
||||
<span class="sd"> :type t: list</span>
|
||||
<span class="sd"> :param `*p`: parameters of the function</span>
|
||||
|
||||
<span class="sd"> * diameter</span>
|
||||
<span class="sd"> * eccentricity</span>
|
||||
<span class="sd"> * torque</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">N</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="n">xppn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">square</span><span class="p">(</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">+</span><span class="n">xn</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span>
|
||||
<span class="p">[</span><span class="n">xppn</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">square</span><span class="p">(</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">+</span><span class="n">xn</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span>
|
||||
<span class="p">[</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xn</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">-</span><span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]]])</span>
|
||||
<span class="n">dN</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">square</span><span class="p">(</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</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="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">square</span><span class="p">(</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
|
||||
<span class="p">[</span><span class="o">-</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xn</span><span class="p">[</span><span class="mi">1</span><span class="p">])]])</span>
|
||||
<span class="n">dNp</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]],</span>
|
||||
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]],</span>
|
||||
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
|
||||
<span class="n">dNpp</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</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="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
|
||||
<span class="p">[</span><span class="mi">0</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="k">return</span> <span class="n">N</span><span class="p">,</span> <span class="n">dN</span><span class="p">,</span> <span class="n">dNp</span><span class="p">,</span> <span class="n">dNpp</span></div>
|
||||
|
||||
<div class="viewcode-block" id="disk_nmmdk"><a class="viewcode-back" href="../../numerical.html#numerical.ode_model.disk_nmmdk">[docs]</a><span class="k">def</span> <span class="nf">disk_nmmdk</span><span class="p">(</span><span class="n">xn</span><span class="p">,</span> <span class="n">xpn</span><span class="p">,</span> <span class="n">xppn</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">):</span>
|
||||
<span class="sd">"""Rotation of an eccentric disk.</span>
|
||||
|
||||
<span class="sd"> :param xn: values of the function</span>
|
||||
<span class="sd"> :type xn: list</span>
|
||||
<span class="sd"> :param xpn: derivative values of the function</span>
|
||||
<span class="sd"> :type xpn: list</span>
|
||||
<span class="sd"> :param xppn: second derivative values of the function</span>
|
||||
<span class="sd"> :type xppn: list</span>
|
||||
<span class="sd"> :param t: time</span>
|
||||
<span class="sd"> :type t: list</span>
|
||||
<span class="sd"> :param `*p`: parameters of the function</span>
|
||||
|
||||
<span class="sd"> * diameter</span>
|
||||
<span class="sd"> * eccentricity</span>
|
||||
<span class="sd"> * torque</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">rm</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="n">xppn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">]],</span>
|
||||
<span class="p">[</span><span class="n">xppn</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">]],</span>
|
||||
<span class="p">[</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">]]])</span>
|
||||
<span class="n">rmx</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
|
||||
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">xppn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
|
||||
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
|
||||
<span class="n">rmxpp</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</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="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
|
||||
<span class="p">[</span><span class="mi">0</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="n">rd</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">square</span><span class="p">(</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
|
||||
<span class="p">[</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">square</span><span class="p">(</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
|
||||
<span class="p">[</span><span class="mi">0</span><span class="p">]])</span>
|
||||
<span class="n">rdx</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">square</span><span class="p">(</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
|
||||
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">square</span><span class="p">(</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span>
|
||||
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
|
||||
<span class="n">rdxp</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]],</span>
|
||||
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xpn</span><span class="p">[</span><span class="mi">2</span><span class="p">]],</span>
|
||||
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
|
||||
<span class="n">rk</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="n">xn</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span>
|
||||
<span class="p">[</span><span class="n">xn</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span>
|
||||
<span class="p">[</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xn</span><span class="p">[</span><span class="mi">1</span><span class="p">])]])</span>
|
||||
<span class="n">rkx</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</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">0</span><span class="p">],</span>
|
||||
<span class="p">[</span><span class="o">-</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">cos</span><span class="p">(</span><span class="n">xn</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">*</span><span class="n">xn</span><span class="p">[</span><span class="mi">1</span><span class="p">])]])</span>
|
||||
<span class="n">f</span> <span class="o">=</span> <span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]]])</span>
|
||||
<span class="k">return</span> <span class="n">rm</span><span class="p">,</span> <span class="n">rmx</span><span class="p">,</span> <span class="n">rmxpp</span><span class="p">,</span> <span class="n">rd</span><span class="p">,</span> <span class="n">rdx</span><span class="p">,</span> <span class="n">rdxp</span><span class="p">,</span> <span class="n">rk</span><span class="p">,</span> <span class="n">rkx</span><span class="p">,</span> <span class="n">f</span></div>
|
||||
|
||||
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">'__main__'</span><span class="p">:</span>
|
||||
<span class="kc">True</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
<div class="sphinxsidebar" role="navigation" aria-label="main navigation">
|
||||
<div class="sphinxsidebarwrapper">
|
||||
<h1 class="logo"><a href="../../index.html">pylib</a></h1>
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
<h3>Navigation</h3>
|
||||
|
||||
<div class="relations">
|
||||
<h3>Related Topics</h3>
|
||||
<ul>
|
||||
<li><a href="../../index.html">Documentation overview</a><ul>
|
||||
<li><a href="../index.html">Module code</a><ul>
|
||||
</ul></li>
|
||||
</ul></li>
|
||||
</ul>
|
||||
</div>
|
||||
<div id="searchbox" style="display: none" role="search">
|
||||
<h3>Quick search</h3>
|
||||
<div class="searchformwrapper">
|
||||
<form class="search" action="../../search.html" method="get">
|
||||
<input type="text" name="q" />
|
||||
<input type="submit" value="Go" />
|
||||
</form>
|
||||
</div>
|
||||
</div>
|
||||
<script type="text/javascript">$('#searchbox').show(0);</script>
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
</div>
|
||||
</div>
|
||||
<div class="clearer"></div>
|
||||
</div>
|
||||
<div class="footer">
|
||||
©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>
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
7
docs/build/html/_sources/fit.rst.txt
vendored
7
docs/build/html/_sources/fit.rst.txt
vendored
@@ -1,7 +0,0 @@
|
||||
fit module
|
||||
==========
|
||||
|
||||
.. automodule:: fit
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
4
docs/build/html/_sources/modules.rst.txt
vendored
4
docs/build/html/_sources/modules.rst.txt
vendored
@@ -6,7 +6,5 @@ src
|
||||
|
||||
data
|
||||
date
|
||||
fit
|
||||
geometry
|
||||
solver
|
||||
solver_model
|
||||
numerical
|
||||
|
||||
46
docs/build/html/_sources/numerical.rst.txt
vendored
Normal file
46
docs/build/html/_sources/numerical.rst.txt
vendored
Normal file
@@ -0,0 +1,46 @@
|
||||
numerical package
|
||||
=================
|
||||
|
||||
Submodules
|
||||
----------
|
||||
|
||||
numerical.fit module
|
||||
--------------------
|
||||
|
||||
.. automodule:: numerical.fit
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
|
||||
numerical.integration module
|
||||
----------------------------
|
||||
|
||||
.. automodule:: numerical.integration
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
|
||||
numerical.ode module
|
||||
--------------------
|
||||
|
||||
.. automodule:: numerical.ode
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
|
||||
numerical.ode\_model module
|
||||
---------------------------
|
||||
|
||||
.. automodule:: numerical.ode_model
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
|
||||
|
||||
Module contents
|
||||
---------------
|
||||
|
||||
.. automodule:: numerical
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
7
docs/build/html/_sources/solver.rst.txt
vendored
7
docs/build/html/_sources/solver.rst.txt
vendored
@@ -1,7 +0,0 @@
|
||||
solver module
|
||||
=============
|
||||
|
||||
.. automodule:: solver
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
@@ -1,7 +0,0 @@
|
||||
solver\_model module
|
||||
====================
|
||||
|
||||
.. automodule:: solver_model
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
6
docs/build/html/_static/alabaster.css
vendored
6
docs/build/html/_static/alabaster.css
vendored
@@ -310,7 +310,7 @@ div.hint {
|
||||
}
|
||||
|
||||
div.seealso {
|
||||
background-color: #3c3c3c;
|
||||
background-color: #25272c;
|
||||
border: 1px solid #2C2C2C;
|
||||
}
|
||||
|
||||
@@ -439,14 +439,14 @@ ul, ol {
|
||||
}
|
||||
|
||||
pre {
|
||||
background: #EEE;
|
||||
background: #25272c;
|
||||
padding: 7px 30px;
|
||||
margin: 15px 0px;
|
||||
line-height: 1.3em;
|
||||
}
|
||||
|
||||
div.viewcode-block:target {
|
||||
background: #ffd;
|
||||
background: #292b2e;
|
||||
}
|
||||
|
||||
dl pre, blockquote pre, li pre {
|
||||
|
||||
6
docs/build/html/_static/custom.css
vendored
6
docs/build/html/_static/custom.css
vendored
@@ -5,3 +5,9 @@
|
||||
table.indextable tr.cap {
|
||||
background-color: #333;
|
||||
}
|
||||
/* move doc link a bit up */
|
||||
.viewcode-back {
|
||||
float: right;
|
||||
position: relative;
|
||||
top: -1.5em;
|
||||
}
|
||||
|
||||
155
docs/build/html/_static/pygments.css
vendored
155
docs/build/html/_static/pygments.css
vendored
@@ -1,77 +1,78 @@
|
||||
.highlight .hll { background-color: #ffffcc }
|
||||
.highlight { background: #f8f8f8; }
|
||||
.highlight .c { color: #8f5902; font-style: italic } /* Comment */
|
||||
.highlight .err { color: #a40000; border: 1px solid #ef2929 } /* Error */
|
||||
.highlight .g { color: #000000 } /* Generic */
|
||||
.highlight .k { color: #004461; font-weight: bold } /* Keyword */
|
||||
.highlight .l { color: #000000 } /* Literal */
|
||||
.highlight .n { color: #000000 } /* Name */
|
||||
.highlight .o { color: #582800 } /* Operator */
|
||||
.highlight .x { color: #000000 } /* Other */
|
||||
.highlight .p { color: #000000; font-weight: bold } /* Punctuation */
|
||||
.highlight .ch { color: #8f5902; font-style: italic } /* Comment.Hashbang */
|
||||
.highlight .cm { color: #8f5902; font-style: italic } /* Comment.Multiline */
|
||||
.highlight .cp { color: #8f5902 } /* Comment.Preproc */
|
||||
.highlight .cpf { color: #8f5902; font-style: italic } /* Comment.PreprocFile */
|
||||
.highlight .c1 { color: #8f5902; font-style: italic } /* Comment.Single */
|
||||
.highlight .cs { color: #8f5902; font-style: italic } /* Comment.Special */
|
||||
.highlight .gd { color: #a40000 } /* Generic.Deleted */
|
||||
.highlight .ge { color: #000000; font-style: italic } /* Generic.Emph */
|
||||
.highlight .gr { color: #ef2929 } /* Generic.Error */
|
||||
.highlight .gh { color: #000080; font-weight: bold } /* Generic.Heading */
|
||||
.highlight .gi { color: #00A000 } /* Generic.Inserted */
|
||||
.highlight .go { color: #888888 } /* Generic.Output */
|
||||
.highlight .gp { color: #745334 } /* Generic.Prompt */
|
||||
.highlight .gs { color: #000000; font-weight: bold } /* Generic.Strong */
|
||||
.highlight .gu { color: #800080; font-weight: bold } /* Generic.Subheading */
|
||||
.highlight .gt { color: #a40000; font-weight: bold } /* Generic.Traceback */
|
||||
.highlight .kc { color: #004461; font-weight: bold } /* Keyword.Constant */
|
||||
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|
||||
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|
||||
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|
||||
.highlight .kr { color: #004461; font-weight: bold } /* Keyword.Reserved */
|
||||
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|
||||
.highlight .ld { color: #000000 } /* Literal.Date */
|
||||
.highlight .m { color: #990000 } /* Literal.Number */
|
||||
.highlight .s { color: #4e9a06 } /* Literal.String */
|
||||
.highlight .na { color: #c4a000 } /* Name.Attribute */
|
||||
.highlight .nb { color: #004461 } /* Name.Builtin */
|
||||
.highlight .nc { color: #000000 } /* Name.Class */
|
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.highlight .no { color: #000000 } /* Name.Constant */
|
||||
.highlight .nd { color: #888888 } /* Name.Decorator */
|
||||
.highlight .ni { color: #ce5c00 } /* Name.Entity */
|
||||
.highlight .ne { color: #cc0000; font-weight: bold } /* Name.Exception */
|
||||
.highlight .nf { color: #000000 } /* Name.Function */
|
||||
.highlight .nl { color: #f57900 } /* Name.Label */
|
||||
.highlight .nn { color: #000000 } /* Name.Namespace */
|
||||
.highlight .nx { color: #000000 } /* Name.Other */
|
||||
.highlight .py { color: #000000 } /* Name.Property */
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8
docs/build/html/data.html
vendored
8
docs/build/html/data.html
vendored
@@ -37,7 +37,7 @@
|
||||
<p>Read and write data to or from file.</p>
|
||||
<span class="target" id="module-data"></span><dl class="function">
|
||||
<dt id="data.data_load">
|
||||
<code class="descname">data_load</code><span class="sig-paren">(</span><em>file_name</em>, <em>verbose=False</em><span class="sig-paren">)</span><a class="headerlink" href="#data.data_load" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">data_load</code><span class="sig-paren">(</span><em>file_name</em>, <em>verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/data.html#data_load"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#data.data_load" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Load stored program objects from binary file.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
@@ -57,7 +57,7 @@
|
||||
|
||||
<dl class="function">
|
||||
<dt id="data.data_read">
|
||||
<code class="descname">data_read</code><span class="sig-paren">(</span><em>file_name</em>, <em>x_column</em>, <em>y_column</em><span class="sig-paren">)</span><a class="headerlink" href="#data.data_read" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">data_read</code><span class="sig-paren">(</span><em>file_name</em>, <em>x_column</em>, <em>y_column</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/data.html#data_read"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#data.data_read" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Read ascii data file.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
@@ -78,7 +78,7 @@
|
||||
|
||||
<dl class="function">
|
||||
<dt id="data.data_store">
|
||||
<code class="descname">data_store</code><span class="sig-paren">(</span><em>file_name</em>, <em>object_data</em><span class="sig-paren">)</span><a class="headerlink" href="#data.data_store" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">data_store</code><span class="sig-paren">(</span><em>file_name</em>, <em>object_data</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/data.html#data_store"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#data.data_store" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Store program objects to binary file.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
@@ -92,7 +92,7 @@
|
||||
|
||||
<dl class="function">
|
||||
<dt id="data.main">
|
||||
<code class="descname">main</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#data.main" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">main</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="_modules/data.html#main"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#data.main" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Main function.</p>
|
||||
</dd></dl>
|
||||
|
||||
|
||||
12
docs/build/html/date.html
vendored
12
docs/build/html/date.html
vendored
@@ -42,7 +42,7 @@
|
||||
</dl>
|
||||
<span class="target" id="module-date"></span><dl class="function">
|
||||
<dt id="date.ascension_of_jesus">
|
||||
<code class="descname">ascension_of_jesus</code><span class="sig-paren">(</span><em>year</em><span class="sig-paren">)</span><a class="headerlink" href="#date.ascension_of_jesus" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">ascension_of_jesus</code><span class="sig-paren">(</span><em>year</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/date.html#ascension_of_jesus"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#date.ascension_of_jesus" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Ascension of Jesus.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
@@ -59,7 +59,7 @@
|
||||
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||||
<dl class="function">
|
||||
<dt id="date.easter_friday">
|
||||
<code class="descname">easter_friday</code><span class="sig-paren">(</span><em>year</em><span class="sig-paren">)</span><a class="headerlink" href="#date.easter_friday" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">easter_friday</code><span class="sig-paren">(</span><em>year</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/date.html#easter_friday"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#date.easter_friday" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Easter Friday.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
@@ -76,7 +76,7 @@
|
||||
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||||
<dl class="function">
|
||||
<dt id="date.easter_monday">
|
||||
<code class="descname">easter_monday</code><span class="sig-paren">(</span><em>year</em><span class="sig-paren">)</span><a class="headerlink" href="#date.easter_monday" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">easter_monday</code><span class="sig-paren">(</span><em>year</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/date.html#easter_monday"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#date.easter_monday" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Easter Monday.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
@@ -93,7 +93,7 @@
|
||||
|
||||
<dl class="function">
|
||||
<dt id="date.easter_sunday">
|
||||
<code class="descname">easter_sunday</code><span class="sig-paren">(</span><em>year</em><span class="sig-paren">)</span><a class="headerlink" href="#date.easter_sunday" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">easter_sunday</code><span class="sig-paren">(</span><em>year</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/date.html#easter_sunday"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#date.easter_sunday" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Easter Sunday.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
@@ -110,7 +110,7 @@
|
||||
|
||||
<dl class="function">
|
||||
<dt id="date.gaußsche_osterformel">
|
||||
<code class="descname">gaußsche_osterformel</code><span class="sig-paren">(</span><em>year</em><span class="sig-paren">)</span><a class="headerlink" href="#date.gaußsche_osterformel" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">gaußsche_osterformel</code><span class="sig-paren">(</span><em>year</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/date.html#gaußsche_osterformel"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#date.gaußsche_osterformel" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Gaußsche Osterformel.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
@@ -144,7 +144,7 @@
|
||||
|
||||
<dl class="function">
|
||||
<dt id="date.pentecost">
|
||||
<code class="descname">pentecost</code><span class="sig-paren">(</span><em>year</em><span class="sig-paren">)</span><a class="headerlink" href="#date.pentecost" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">pentecost</code><span class="sig-paren">(</span><em>year</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/date.html#pentecost"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#date.pentecost" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Pentecost.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
|
||||
173
docs/build/html/fit.html
vendored
173
docs/build/html/fit.html
vendored
@@ -1,173 +0,0 @@
|
||||
|
||||
<!DOCTYPE html>
|
||||
|
||||
<html xmlns="http://www.w3.org/1999/xhtml">
|
||||
<head>
|
||||
<meta charset="utf-8" />
|
||||
<title>fit module — pylib 2019.5.19 documentation</title>
|
||||
<link rel="stylesheet" href="_static/alabaster.css" type="text/css" />
|
||||
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
|
||||
<link rel="stylesheet" href="_static/custom.css" type="text/css" />
|
||||
<script type="text/javascript" id="documentation_options" data-url_root="./" src="_static/documentation_options.js"></script>
|
||||
<script type="text/javascript" src="_static/jquery.js"></script>
|
||||
<script type="text/javascript" src="_static/underscore.js"></script>
|
||||
<script type="text/javascript" src="_static/doctools.js"></script>
|
||||
<script type="text/javascript" src="_static/language_data.js"></script>
|
||||
<script async="async" type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-AMS-MML_HTMLorMML"></script>
|
||||
<link rel="index" title="Index" href="genindex.html" />
|
||||
<link rel="search" title="Search" href="search.html" />
|
||||
|
||||
<link rel="stylesheet" href="_static/custom.css" type="text/css" />
|
||||
|
||||
|
||||
<meta name="viewport" content="width=device-width, initial-scale=0.9, maximum-scale=0.9" />
|
||||
|
||||
</head><body>
|
||||
|
||||
|
||||
<div class="document">
|
||||
<div class="documentwrapper">
|
||||
<div class="bodywrapper">
|
||||
|
||||
|
||||
<div class="body" role="main">
|
||||
|
||||
<div class="section" id="module-fit">
|
||||
<span id="fit-module"></span><h1>fit module<a class="headerlink" href="#module-fit" title="Permalink to this headline">¶</a></h1>
|
||||
<p>Function and approximation.</p>
|
||||
<span class="target" id="module-fit"></span><dl class="function">
|
||||
<dt id="fit.gauss">
|
||||
<code class="descname">gauss</code><span class="sig-paren">(</span><em>x</em>, <em>*p</em><span class="sig-paren">)</span><a class="headerlink" href="#fit.gauss" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Gauss distribution function.</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[f(x)=ae^{-(x-b)^{2}/(2c^{2})}\]</div>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
<dd class="field-odd"><ul class="simple">
|
||||
<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>
|
||||
<li><p><strong>p</strong> (<em>list</em>) – <p>gauss parameters [a, b, c, d]:</p>
|
||||
<ul>
|
||||
<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>
|
||||
<li><p>b – expected value <span class="math notranslate nohighlight">\(\mu\)</span> (position of maximum, default = 0)</p></li>
|
||||
<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>
|
||||
<li><p>d – vertical offset (default = 0)</p></li>
|
||||
</ul>
|
||||
</p></li>
|
||||
</ul>
|
||||
</dd>
|
||||
<dt class="field-even">Returns</dt>
|
||||
<dd class="field-even"><p>gauss values at given positions x</p>
|
||||
</dd>
|
||||
<dt class="field-odd">Return type</dt>
|
||||
<dd class="field-odd"><p>numpy.ndarray</p>
|
||||
</dd>
|
||||
</dl>
|
||||
</dd></dl>
|
||||
|
||||
<dl class="function">
|
||||
<dt id="fit.gauss_fit">
|
||||
<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="headerlink" href="#fit.gauss_fit" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Fit Gauss distribution function to data.</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>int</em><em> or </em><em>float</em><em> or </em><em>list</em><em> or </em><em>numpy.ndarray</em>) – positions</p></li>
|
||||
<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>
|
||||
<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>
|
||||
<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
|
||||
is used)</p></li>
|
||||
<li><p><strong>verbose</strong> (<em>bool</em>) – verbose information (default = False)</p></li>
|
||||
</ul>
|
||||
</dd>
|
||||
<dt class="field-even">Returns</dt>
|
||||
<dd class="field-even"><p><ul class="simple">
|
||||
<li><p>numpy.ndarray – fitted values (y_fit)</p></li>
|
||||
<li><p>numpy.ndarray – parameters of gauss distribution function (popt:
|
||||
amplitude a, expected value <span class="math notranslate nohighlight">\(\mu\)</span>, standard deviation
|
||||
<span class="math notranslate nohighlight">\(\sigma\)</span>, vertical offset d)</p></li>
|
||||
<li><p>numpy.float64 – full width at half maximum (FWHM)</p></li>
|
||||
</ul>
|
||||
</p>
|
||||
</dd>
|
||||
<dt class="field-odd">Return type</dt>
|
||||
<dd class="field-odd"><p>tuple</p>
|
||||
</dd>
|
||||
</dl>
|
||||
<div class="admonition seealso">
|
||||
<p class="admonition-title">See also</p>
|
||||
<p><a class="reference internal" href="#fit.gauss" title="fit.gauss"><code class="xref py py-meth docutils literal notranslate"><span class="pre">gauss()</span></code></a></p>
|
||||
</div>
|
||||
</dd></dl>
|
||||
|
||||
<dl class="function">
|
||||
<dt id="fit.main">
|
||||
<code class="descname">main</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#fit.main" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>test function</p>
|
||||
</dd></dl>
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
<div class="sphinxsidebar" role="navigation" aria-label="main navigation">
|
||||
<div class="sphinxsidebarwrapper">
|
||||
<h1 class="logo"><a href="index.html">pylib</a></h1>
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
<h3>Navigation</h3>
|
||||
|
||||
<div class="relations">
|
||||
<h3>Related Topics</h3>
|
||||
<ul>
|
||||
<li><a href="index.html">Documentation overview</a><ul>
|
||||
</ul></li>
|
||||
</ul>
|
||||
</div>
|
||||
<div id="searchbox" style="display: none" role="search">
|
||||
<h3>Quick search</h3>
|
||||
<div class="searchformwrapper">
|
||||
<form class="search" action="search.html" method="get">
|
||||
<input type="text" name="q" />
|
||||
<input type="submit" value="Go" />
|
||||
</form>
|
||||
</div>
|
||||
</div>
|
||||
<script type="text/javascript">$('#searchbox').show(0);</script>
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
</div>
|
||||
</div>
|
||||
<div class="clearer"></div>
|
||||
</div>
|
||||
<div class="footer">
|
||||
©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/fit.rst.txt"
|
||||
rel="nofollow">Page source</a>
|
||||
</div>
|
||||
|
||||
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
73
docs/build/html/genindex.html
vendored
73
docs/build/html/genindex.html
vendored
@@ -47,6 +47,7 @@
|
||||
| <a href="#L"><strong>L</strong></a>
|
||||
| <a href="#M"><strong>M</strong></a>
|
||||
| <a href="#N"><strong>N</strong></a>
|
||||
| <a href="#O"><strong>O</strong></a>
|
||||
| <a href="#P"><strong>P</strong></a>
|
||||
| <a href="#R"><strong>R</strong></a>
|
||||
| <a href="#S"><strong>S</strong></a>
|
||||
@@ -88,11 +89,13 @@
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
<li><a href="date.html#module-date">date (module)</a>, <a href="date.html#module-date">[1]</a>
|
||||
</li>
|
||||
<li><a href="solver_model.html#solver_model.disk">disk() (in module solver_model)</a>
|
||||
<li><a href="numerical.html#numerical.ode_model.disk">disk() (in module numerical.ode_model)</a>
|
||||
</li>
|
||||
<li><a href="solver_model.html#solver_model.disk_nm">disk_nm() (in module solver_model)</a>
|
||||
<li><a href="numerical.html#numerical.ode_model.disk_nm">disk_nm() (in module numerical.ode_model)</a>
|
||||
</li>
|
||||
<li><a href="solver_model.html#solver_model.disk_nmmdk">disk_nmmdk() (in module solver_model)</a>
|
||||
<li><a href="numerical.html#numerical.ode_model.disk_nmmdk">disk_nmmdk() (in module numerical.ode_model)</a>
|
||||
</li>
|
||||
<li><a href="numerical.html#numerical.ode.dxdt_Dt">dxdt_Dt() (in module numerical.ode)</a>
|
||||
</li>
|
||||
</ul></td>
|
||||
</tr></table>
|
||||
@@ -100,11 +103,11 @@
|
||||
<h2 id="E">E</h2>
|
||||
<table style="width: 100%" class="indextable genindextable"><tr>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
<li><a href="solver.html#solver.e1">e1() (in module solver)</a>
|
||||
<li><a href="numerical.html#numerical.ode.e1">e1() (in module numerical.ode)</a>
|
||||
</li>
|
||||
<li><a href="solver.html#solver.e2">e2() (in module solver)</a>
|
||||
<li><a href="numerical.html#numerical.ode.e2">e2() (in module numerical.ode)</a>
|
||||
</li>
|
||||
<li><a href="solver.html#solver.e4">e4() (in module solver)</a>
|
||||
<li><a href="numerical.html#numerical.ode.e4">e4() (in module numerical.ode)</a>
|
||||
</li>
|
||||
</ul></td>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
@@ -120,7 +123,11 @@
|
||||
<h2 id="F">F</h2>
|
||||
<table style="width: 100%" class="indextable genindextable"><tr>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
<li><a href="fit.html#module-fit">fit (module)</a>, <a href="fit.html#module-fit">[1]</a>
|
||||
<li><a href="numerical.html#module-fit">fit (module)</a>
|
||||
</li>
|
||||
</ul></td>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
<li><a href="numerical.html#numerical.ode.fixed_point_iteration">fixed_point_iteration() (in module numerical.ode)</a>
|
||||
</li>
|
||||
</ul></td>
|
||||
</tr></table>
|
||||
@@ -128,9 +135,9 @@
|
||||
<h2 id="G">G</h2>
|
||||
<table style="width: 100%" class="indextable genindextable"><tr>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
<li><a href="fit.html#fit.gauss">gauss() (in module fit)</a>
|
||||
<li><a href="numerical.html#numerical.fit.gauss">gauss() (in module numerical.fit)</a>
|
||||
</li>
|
||||
<li><a href="fit.html#fit.gauss_fit">gauss_fit() (in module fit)</a>
|
||||
<li><a href="numerical.html#numerical.fit.gauss_fit">gauss_fit() (in module numerical.fit)</a>
|
||||
</li>
|
||||
</ul></td>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
@@ -144,7 +151,13 @@
|
||||
<h2 id="I">I</h2>
|
||||
<table style="width: 100%" class="indextable genindextable"><tr>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
<li><a href="solver.html#solver.i1">i1() (in module solver)</a>
|
||||
<li><a href="numerical.html#numerical.ode.i1">i1() (in module numerical.ode)</a>
|
||||
</li>
|
||||
</ul></td>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
<li><a href="numerical.html#numerical.ode.i1n">i1n() (in module numerical.ode)</a>
|
||||
</li>
|
||||
<li><a href="numerical.html#module-integration">integration (module)</a>
|
||||
</li>
|
||||
</ul></td>
|
||||
</tr></table>
|
||||
@@ -161,22 +174,40 @@
|
||||
<table style="width: 100%" class="indextable genindextable"><tr>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
<li><a href="data.html#data.main">main() (in module data)</a>
|
||||
|
||||
<ul>
|
||||
<li><a href="fit.html#fit.main">(in module fit)</a>
|
||||
</li>
|
||||
</ul></li>
|
||||
</ul></td>
|
||||
</tr></table>
|
||||
|
||||
<h2 id="N">N</h2>
|
||||
<table style="width: 100%" class="indextable genindextable"><tr>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
<li><a href="solver.html#solver.newmark_newtonraphson">newmark_newtonraphson() (in module solver)</a>
|
||||
<li><a href="numerical.html#numerical.ode.newmark_newtonraphson">newmark_newtonraphson() (in module numerical.ode)</a>
|
||||
</li>
|
||||
<li><a href="numerical.html#numerical.ode.newmark_newtonraphson_rdk">newmark_newtonraphson_rdk() (in module numerical.ode)</a>
|
||||
</li>
|
||||
<li><a href="numerical.html#module-numerical">numerical (module)</a>
|
||||
</li>
|
||||
</ul></td>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
<li><a href="solver.html#solver.newmark_newtonraphson_rdk">newmark_newtonraphson_rdk() (in module solver)</a>
|
||||
<li><a href="numerical.html#module-numerical.fit">numerical.fit (module)</a>
|
||||
</li>
|
||||
<li><a href="numerical.html#module-numerical.integration">numerical.integration (module)</a>
|
||||
</li>
|
||||
<li><a href="numerical.html#module-numerical.ode">numerical.ode (module)</a>
|
||||
</li>
|
||||
<li><a href="numerical.html#module-numerical.ode_model">numerical.ode_model (module)</a>
|
||||
</li>
|
||||
</ul></td>
|
||||
</tr></table>
|
||||
|
||||
<h2 id="O">O</h2>
|
||||
<table style="width: 100%" class="indextable genindextable"><tr>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
<li><a href="numerical.html#module-ode">ode (module)</a>
|
||||
</li>
|
||||
</ul></td>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
<li><a href="numerical.html#module-ode_model">ode_model (module)</a>
|
||||
</li>
|
||||
</ul></td>
|
||||
</tr></table>
|
||||
@@ -210,12 +241,6 @@
|
||||
<h2 id="S">S</h2>
|
||||
<table style="width: 100%" class="indextable genindextable"><tr>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
<li><a href="solver.html#module-solver">solver (module)</a>, <a href="solver.html#module-solver">[1]</a>
|
||||
</li>
|
||||
</ul></td>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
<li><a href="solver_model.html#module-solver_model">solver_model (module)</a>, <a href="solver_model.html#module-solver_model">[1]</a>
|
||||
</li>
|
||||
<li><a href="geometry.html#geometry.square">square() (in module geometry)</a>
|
||||
</li>
|
||||
</ul></td>
|
||||
@@ -225,6 +250,10 @@
|
||||
<table style="width: 100%" class="indextable genindextable"><tr>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
<li><a href="geometry.html#geometry.translate">translate() (in module geometry)</a>
|
||||
</li>
|
||||
</ul></td>
|
||||
<td style="width: 33%; vertical-align: top;"><ul>
|
||||
<li><a href="numerical.html#numerical.integration.trapez">trapez() (in module numerical.integration)</a>
|
||||
</li>
|
||||
</ul></td>
|
||||
</tr></table>
|
||||
|
||||
18
docs/build/html/geometry.html
vendored
18
docs/build/html/geometry.html
vendored
@@ -42,7 +42,7 @@
|
||||
</dl>
|
||||
<span class="target" id="module-geometry"></span><dl class="function">
|
||||
<dt id="geometry.cubic">
|
||||
<code class="descname">cubic</code><span class="sig-paren">(</span><em>point1</em>, <em>angle1</em>, <em>point2</em>, <em>angle2</em>, <em>samples=50</em><span class="sig-paren">)</span><a class="headerlink" href="#geometry.cubic" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">cubic</code><span class="sig-paren">(</span><em>point1</em>, <em>angle1</em>, <em>point2</em>, <em>angle2</em>, <em>samples=50</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/geometry.html#cubic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#geometry.cubic" 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">
|
||||
@@ -65,7 +65,7 @@
|
||||
|
||||
<dl class="function">
|
||||
<dt id="geometry.cubic_deg">
|
||||
<code class="descname">cubic_deg</code><span class="sig-paren">(</span><em>point1</em>, <em>angle1</em>, <em>point2</em>, <em>angle2</em><span class="sig-paren">)</span><a class="headerlink" href="#geometry.cubic_deg" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">cubic_deg</code><span class="sig-paren">(</span><em>point1</em>, <em>angle1</em>, <em>point2</em>, <em>angle2</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/geometry.html#cubic_deg"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#geometry.cubic_deg" 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">
|
||||
@@ -91,7 +91,7 @@
|
||||
|
||||
<dl class="function">
|
||||
<dt id="geometry.line">
|
||||
<code class="descname">line</code><span class="sig-paren">(</span><em>point1</em>, <em>point2</em>, <em>samples=2</em><span class="sig-paren">)</span><a class="headerlink" href="#geometry.line" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">line</code><span class="sig-paren">(</span><em>point1</em>, <em>point2</em>, <em>samples=2</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/geometry.html#line"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#geometry.line" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><div class="math notranslate nohighlight">
|
||||
\[y = \frac{y_2-y_1}{x_2-x_1}(x-x_1) + y_1\]</div>
|
||||
<dl class="field-list simple">
|
||||
@@ -115,7 +115,7 @@
|
||||
|
||||
<dl class="function">
|
||||
<dt id="geometry.points">
|
||||
<code class="descname">points</code><span class="sig-paren">(</span><em>*pts</em><span class="sig-paren">)</span><a class="headerlink" href="#geometry.points" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">points</code><span class="sig-paren">(</span><em>*pts</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/geometry.html#points"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#geometry.points" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
<dd class="field-odd"><p><strong>*pts</strong> – points to rearrange</p>
|
||||
@@ -131,7 +131,7 @@
|
||||
|
||||
<dl class="function">
|
||||
<dt id="geometry.rectangle">
|
||||
<code class="descname">rectangle</code><span class="sig-paren">(</span><em>width</em>, <em>height</em><span class="sig-paren">)</span><a class="headerlink" href="#geometry.rectangle" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">rectangle</code><span class="sig-paren">(</span><em>width</em>, <em>height</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/geometry.html#rectangle"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#geometry.rectangle" 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">
|
||||
@@ -150,7 +150,7 @@
|
||||
|
||||
<dl class="function">
|
||||
<dt id="geometry.rotate">
|
||||
<code class="descname">rotate</code><span class="sig-paren">(</span><em>origin</em>, <em>angle</em>, <em>*pts</em>, <em>**kwargs</em><span class="sig-paren">)</span><a class="headerlink" href="#geometry.rotate" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">rotate</code><span class="sig-paren">(</span><em>origin</em>, <em>angle</em>, <em>*pts</em>, <em>**kwargs</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/geometry.html#rotate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#geometry.rotate" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Rotate a point or polygon counterclockwise by a given angle around a given
|
||||
origin. The angle should be given in radians.</p>
|
||||
<dl class="field-list simple">
|
||||
@@ -173,7 +173,7 @@ origin. The angle should be given in radians.</p>
|
||||
|
||||
<dl class="function">
|
||||
<dt id="geometry.rotate_deg">
|
||||
<code class="descname">rotate_deg</code><span class="sig-paren">(</span><em>origin</em>, <em>angle</em>, <em>*pts</em>, <em>**kwargs</em><span class="sig-paren">)</span><a class="headerlink" href="#geometry.rotate_deg" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">rotate_deg</code><span class="sig-paren">(</span><em>origin</em>, <em>angle</em>, <em>*pts</em>, <em>**kwargs</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/geometry.html#rotate_deg"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#geometry.rotate_deg" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Rotate a point or polygon counterclockwise by a given angle around a given
|
||||
origin. The angle should be given in degrees.</p>
|
||||
<dl class="field-list simple">
|
||||
@@ -200,7 +200,7 @@ origin. The angle should be given in degrees.</p>
|
||||
|
||||
<dl class="function">
|
||||
<dt id="geometry.square">
|
||||
<code class="descname">square</code><span class="sig-paren">(</span><em>width</em><span class="sig-paren">)</span><a class="headerlink" href="#geometry.square" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">square</code><span class="sig-paren">(</span><em>width</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/geometry.html#square"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#geometry.square" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
<dd class="field-odd"><p><strong>width</strong> (<em>int</em><em> or </em><em>float</em>) – the edge size of the square</p>
|
||||
@@ -220,7 +220,7 @@ origin. The angle should be given in degrees.</p>
|
||||
|
||||
<dl class="function">
|
||||
<dt id="geometry.translate">
|
||||
<code class="descname">translate</code><span class="sig-paren">(</span><em>vec</em>, <em>*pts</em><span class="sig-paren">)</span><a class="headerlink" href="#geometry.translate" title="Permalink to this definition">¶</a></dt>
|
||||
<code class="descname">translate</code><span class="sig-paren">(</span><em>vec</em>, <em>*pts</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/geometry.html#translate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#geometry.translate" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Translate a point or polygon by a given vector.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
|
||||
12
docs/build/html/modules.html
vendored
12
docs/build/html/modules.html
vendored
@@ -38,10 +38,16 @@
|
||||
<ul>
|
||||
<li class="toctree-l1"><a class="reference internal" href="data.html">data module</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="date.html">date module</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="fit.html">fit module</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="geometry.html">geometry module</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="solver.html">solver module</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="solver_model.html">solver_model module</a></li>
|
||||
<li class="toctree-l1"><a class="reference internal" href="numerical.html">numerical package</a><ul>
|
||||
<li class="toctree-l2"><a class="reference internal" href="numerical.html#submodules">Submodules</a></li>
|
||||
<li class="toctree-l2"><a class="reference internal" href="numerical.html#module-numerical.fit">numerical.fit module</a></li>
|
||||
<li class="toctree-l2"><a class="reference internal" href="numerical.html#module-numerical.integration">numerical.integration module</a></li>
|
||||
<li class="toctree-l2"><a class="reference internal" href="numerical.html#module-numerical.ode">numerical.ode module</a></li>
|
||||
<li class="toctree-l2"><a class="reference internal" href="numerical.html#module-numerical.ode_model">numerical.ode_model module</a></li>
|
||||
<li class="toctree-l2"><a class="reference internal" href="numerical.html#module-numerical">Module contents</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
586
docs/build/html/numerical.html
vendored
Normal file
586
docs/build/html/numerical.html
vendored
Normal file
@@ -0,0 +1,586 @@
|
||||
|
||||
<!DOCTYPE html>
|
||||
|
||||
<html xmlns="http://www.w3.org/1999/xhtml">
|
||||
<head>
|
||||
<meta charset="utf-8" />
|
||||
<title>numerical package — pylib 2019.5.19 documentation</title>
|
||||
<link rel="stylesheet" href="_static/alabaster.css" type="text/css" />
|
||||
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
|
||||
<link rel="stylesheet" href="_static/custom.css" type="text/css" />
|
||||
<script type="text/javascript" id="documentation_options" data-url_root="./" src="_static/documentation_options.js"></script>
|
||||
<script type="text/javascript" src="_static/jquery.js"></script>
|
||||
<script type="text/javascript" src="_static/underscore.js"></script>
|
||||
<script type="text/javascript" src="_static/doctools.js"></script>
|
||||
<script type="text/javascript" src="_static/language_data.js"></script>
|
||||
<script async="async" type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-AMS-MML_HTMLorMML"></script>
|
||||
<link rel="index" title="Index" href="genindex.html" />
|
||||
<link rel="search" title="Search" href="search.html" />
|
||||
|
||||
<link rel="stylesheet" href="_static/custom.css" type="text/css" />
|
||||
|
||||
|
||||
<meta name="viewport" content="width=device-width, initial-scale=0.9, maximum-scale=0.9" />
|
||||
|
||||
</head><body>
|
||||
|
||||
|
||||
<div class="document">
|
||||
<div class="documentwrapper">
|
||||
<div class="bodywrapper">
|
||||
|
||||
|
||||
<div class="body" role="main">
|
||||
|
||||
<div class="section" id="numerical-package">
|
||||
<h1>numerical package<a class="headerlink" href="#numerical-package" title="Permalink to this headline">¶</a></h1>
|
||||
<div class="section" id="submodules">
|
||||
<h2>Submodules<a class="headerlink" href="#submodules" title="Permalink to this headline">¶</a></h2>
|
||||
</div>
|
||||
<div class="section" id="module-numerical.fit">
|
||||
<span id="numerical-fit-module"></span><h2>numerical.fit module<a class="headerlink" href="#module-numerical.fit" title="Permalink to this headline">¶</a></h2>
|
||||
<p>Function and approximation.</p>
|
||||
<span class="target" id="module-fit"></span><dl class="function">
|
||||
<dt id="numerical.fit.gauss">
|
||||
<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>
|
||||
<dd><p>Gauss distribution function.</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[f(x)=ae^{-(x-b)^{2}/(2c^{2})}\]</div>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Parameters</dt>
|
||||
<dd class="field-odd"><ul class="simple">
|
||||
<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>
|
||||
<li><p><strong>p</strong> (<em>list</em>) – <p>gauss parameters [a, b, c, d]:</p>
|
||||
<ul>
|
||||
<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>
|
||||
<li><p>b – expected value <span class="math notranslate nohighlight">\(\mu\)</span> (position of maximum, default = 0)</p></li>
|
||||
<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>
|
||||
<li><p>d – vertical offset (default = 0)</p></li>
|
||||
</ul>
|
||||
</p></li>
|
||||
</ul>
|
||||
</dd>
|
||||
<dt class="field-even">Returns</dt>
|
||||
<dd class="field-even"><p>gauss values at given positions x</p>
|
||||
</dd>
|
||||
<dt class="field-odd">Return type</dt>
|
||||
<dd class="field-odd"><p>numpy.ndarray</p>
|
||||
</dd>
|
||||
</dl>
|
||||
</dd></dl>
|
||||
|
||||
<dl class="function">
|
||||
<dt id="numerical.fit.gauss_fit">
|
||||
<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>
|
||||
<dd><p>Fit Gauss distribution function to data.</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>int</em><em> or </em><em>float</em><em> or </em><em>list</em><em> or </em><em>numpy.ndarray</em>) – positions</p></li>
|
||||
<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>
|
||||
<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>
|
||||
<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
|
||||
is used)</p></li>
|
||||
<li><p><strong>verbose</strong> (<em>bool</em>) – verbose information (default = False)</p></li>
|
||||
</ul>
|
||||
</dd>
|
||||
<dt class="field-even">Returns</dt>
|
||||
<dd class="field-even"><p><ul class="simple">
|
||||
<li><p>numpy.ndarray – fitted values (y_fit)</p></li>
|
||||
<li><p>numpy.ndarray – parameters of gauss distribution function (popt:
|
||||
amplitude a, expected value <span class="math notranslate nohighlight">\(\mu\)</span>, standard deviation
|
||||
<span class="math notranslate nohighlight">\(\sigma\)</span>, vertical offset d)</p></li>
|
||||
<li><p>numpy.float64 – full width at half maximum (FWHM)</p></li>
|
||||
</ul>
|
||||
</p>
|
||||
</dd>
|
||||
<dt class="field-odd">Return type</dt>
|
||||
<dd class="field-odd"><p>tuple</p>
|
||||
</dd>
|
||||
</dl>
|
||||
<div class="admonition seealso">
|
||||
<p class="admonition-title">See also</p>
|
||||
<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>
|
||||
</div>
|
||||
</dd></dl>
|
||||
|
||||
</div>
|
||||
<div class="section" id="module-numerical.integration">
|
||||
<span id="numerical-integration-module"></span><h2>numerical.integration module<a class="headerlink" href="#module-numerical.integration" title="Permalink to this headline">¶</a></h2>
|
||||
<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="numerical.integration.trapez">
|
||||
<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>
|
||||
<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">
|
||||
\[\begin{split}I &= \int\limits_a^b f(x) \,\mathrm{d}x \\
|
||||
I &\approx \int\limits_a^b f_n(x) \,\mathrm{d}x \\
|
||||
&= \int\limits_a^b
|
||||
\left( f(a)+\frac{f(b)-f(a)}{b-a}(x-a) \right)
|
||||
\mathrm{d}x \\
|
||||
&= \left.\left( f(a)-a\frac{f(b)-f(a)}{b-a} \right)
|
||||
x \right\vert_a^b +
|
||||
\left. \frac{f(b)-f(a)}{b-a} \frac{x^2}{2}
|
||||
\right\vert_a^b \\
|
||||
&= \frac{b-a}{2}\left[f(a)+f(b)\right]\end{split}\]</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">
|
||||
\[\begin{split}I &\approx \sum\limits_{i=0}^{n-1} \frac{1}{2} (x_{i+1}-x_i)
|
||||
\left[f(x_{i+1})+f(x_i)\right] \\\end{split}\]</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">
|
||||
\[\begin{split}I &= \int\limits_a^b f(x) \,\mathrm{d}x \\
|
||||
f(x) &= x^2 \\
|
||||
a &= 0 \\
|
||||
b &= 1\end{split}\]</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">>>> </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">>>> </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">>>> </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">>>> </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>
|
||||
|
||||
</div>
|
||||
<div class="section" id="module-numerical.ode">
|
||||
<span id="numerical-ode-module"></span><h2>numerical.ode module<a class="headerlink" href="#module-numerical.ode" title="Permalink to this headline">¶</a></h2>
|
||||
<p>Numerical solver of ordinary differential equations.</p>
|
||||
<p>Solves the initial value problem for systems of first order ordinary differential
|
||||
equations.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Date</dt>
|
||||
<dd class="field-odd"><p>2015-09-21</p>
|
||||
</dd>
|
||||
</dl>
|
||||
<span class="target" id="module-ode"></span><dl class="function">
|
||||
<dt id="numerical.ode.dxdt_Dt">
|
||||
<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>
|
||||
<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>) – <span class="math notranslate nohighlight">\(f = \dot{x}\)</span></p></li>
|
||||
<li><p><strong>Dt</strong> – <span class="math notranslate nohighlight">\(\Delta{t}\)</span></p></li>
|
||||
</ul>
|
||||
</dd>
|
||||
<dt class="field-even">Returns</dt>
|
||||
<dd class="field-even"><p><span class="math notranslate nohighlight">\(\Delta x = \dot{x} \Delta t\)</span></p>
|
||||
</dd>
|
||||
</dl>
|
||||
</dd></dl>
|
||||
|
||||
<dl class="function">
|
||||
<dt id="numerical.ode.e1">
|
||||
<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>
|
||||
<dd><p>Explicit first-order method /
|
||||
(standard, or forward) Euler method /
|
||||
Runge-Kutta 1st order method.</p>
|
||||
<p>de:
|
||||
Euler’sche Polygonzugverfahren / explizite Euler-Verfahren /
|
||||
Euler-Cauchy-Verfahren / Euler-vorwärts-Verfahren</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>
|
||||
<p>Approximate the solution of the initial value problem</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[\begin{split}\dot{x} &= f(t,x) \\
|
||||
x(t_0) &= x_0\end{split}\]</div>
|
||||
<p>Choose a value h for the size of every step and set</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[t_i = t_0 + i h ~,\quad i=1,2,\ldots,n\]</div>
|
||||
<p>The derivative of the solution is approximated as the forward difference
|
||||
equation</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[\dot{x}_i = f(t_i, x_i) = \frac{x_{i+1} - x_i}{t_{i+1}-t_i}\]</div>
|
||||
<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
|
||||
<span class="math notranslate nohighlight">\(t_{i+1}\)</span> is</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[\begin{split}x_{i+1} &= x_i + (t_{i+1}-t_i) f(t_i, x_i) \\
|
||||
x_{i+1} &= x_i + h f(t_i, x_i) \\\end{split}\]</div>
|
||||
<p>Example 1:</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[\begin{split}m\ddot{u} + d\dot{u} + ku = f(t) \\
|
||||
\ddot{u} = m^{-1}(f(t) - d\dot{u} - ku) \\\end{split}\]</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">
|
||||
\[\begin{split}\dot{x}_1 &= x_2 \\
|
||||
\dot{x}_2 &= m^{-1}(f(t) - d x_2 - k x_1) \\\end{split}\]</div>
|
||||
<p>or</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[\begin{split}\dot{x} &= f(t,x) \\
|
||||
\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &=
|
||||
\begin{bmatrix} x_2 \\ m^{-1}(f(t) - d x_2 - k x_1) \end{bmatrix} \\
|
||||
&=
|
||||
\begin{bmatrix} 0 \\ m^{-1} f(t) \end{bmatrix} +
|
||||
\begin{bmatrix} 0 & 1 \\ -m^{-1} k & -m^{-1} d \end{bmatrix}
|
||||
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\end{split}\]</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>
|
||||
|
||||
|
||||
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|
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<a href="_sources/numerical.rst.txt"
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</body>
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</html>
|
||||
10
docs/build/html/objects.inv
vendored
10
docs/build/html/objects.inv
vendored
@@ -2,10 +2,6 @@
|
||||
# Project: pylib
|
||||
# Version:
|
||||
# The remainder of this file is compressed using zlib.
|
||||
xڥ<EFBFBD>Mr<EFBFBD> <10><>s
|
||||
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|
||||
5<EFBFBD>H<EFBFBD>;<07>U<><55>ȷ<EFBFBD><C8B7>^ZCn2E<32>:<3A>F<EFBFBD>[<07>Yͤ<59>F<EFBFBD>|7<><37><1B>3<EFBFBD>`0<>P<EFBFBD>ү<EFBFBD>.<2E><>t<EFBFBD>C<0F><>N
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
xڭ<EFBFBD><EFBFBD><EFBFBD><EFBFBD>0<14><><<3C>h<>! <0C><>4<14>0CA<43>Q<EFBFBD>cGD<17># <15>A<EFBFBD><41><EFBFBD>&< <09><>d<EFBFBD>d<EFBFBD>H<EFBFBD>4<EFBFBD>}<7D><><EFBFBD>f<EFBFBD><04><>L<EFBFBD>;mŬ<6D><C5AC>&"<18><><EFBFBD><EFBFBD>e2u<32>H<EFBFBD>Z<EFBFBD><5A><EFBFBD>*<2A>D<EFBFBD><44><EFBFBD>p/<2F>!<21>LQ<4C><0E>Q<EFBFBD><51><EFBFBD>mV3inQpY
|
||||
<T
|
||||
<EFBFBD> 9 N<> <1D><17><19>=C<><43><07> <1D><14>P<EFBFBD>kkZp<5A><70><EFBFBD><EFBFBD>Ϳ!<21><02>Q7X<37>AݖM`<<3C> <20>':H_<48><5F><EFBFBD><1A><>L]5s<04><><EFBFBD>C<EFBFBD><43><EFBFBD><EFBFBD><EFBFBD>=<3D>7<EFBFBD>_<EFBFBD>P<EFBFBD><50>q*`<60><>(i<><69><EFBFBD><EFBFBD>4 <1E>y<07>33<33><33><10><><EFBFBD><EFBFBD>l<EFBFBD>ۊƯ3s
|
||||
59
docs/build/html/py-modindex.html
vendored
59
docs/build/html/py-modindex.html
vendored
@@ -46,7 +46,9 @@
|
||||
<a href="#cap-d"><strong>d</strong></a> |
|
||||
<a href="#cap-f"><strong>f</strong></a> |
|
||||
<a href="#cap-g"><strong>g</strong></a> |
|
||||
<a href="#cap-s"><strong>s</strong></a>
|
||||
<a href="#cap-i"><strong>i</strong></a> |
|
||||
<a href="#cap-n"><strong>n</strong></a> |
|
||||
<a href="#cap-o"><strong>o</strong></a>
|
||||
</div>
|
||||
|
||||
<table class="indextable modindextable">
|
||||
@@ -69,7 +71,7 @@
|
||||
<tr>
|
||||
<td></td>
|
||||
<td>
|
||||
<a href="fit.html#module-fit"><code class="xref">fit</code></a> <em>(*nix, Windows)</em></td><td>
|
||||
<a href="numerical.html#module-fit"><code class="xref">fit</code></a> <em>(*nix, Windows)</em></td><td>
|
||||
<em>Function and approximation.</em></td></tr>
|
||||
<tr class="pcap"><td></td><td> </td><td></td></tr>
|
||||
<tr class="cap" id="cap-g"><td></td><td>
|
||||
@@ -80,18 +82,61 @@
|
||||
<a href="geometry.html#module-geometry"><code class="xref">geometry</code></a> <em>(*nix, Windows)</em></td><td>
|
||||
<em>Geometry objects.</em></td></tr>
|
||||
<tr class="pcap"><td></td><td> </td><td></td></tr>
|
||||
<tr class="cap" id="cap-s"><td></td><td>
|
||||
<strong>s</strong></td><td></td></tr>
|
||||
<tr class="cap" id="cap-i"><td></td><td>
|
||||
<strong>i</strong></td><td></td></tr>
|
||||
<tr>
|
||||
<td></td>
|
||||
<td>
|
||||
<a href="solver.html#module-solver"><code class="xref">solver</code></a> <em>(*nix, Windows)</em></td><td>
|
||||
<a href="numerical.html#module-integration"><code class="xref">integration</code></a> <em>(*nix, Windows)</em></td><td>
|
||||
<em>Numerical integration.</em></td></tr>
|
||||
<tr class="pcap"><td></td><td> </td><td></td></tr>
|
||||
<tr class="cap" id="cap-n"><td></td><td>
|
||||
<strong>n</strong></td><td></td></tr>
|
||||
<tr>
|
||||
<td><img src="_static/minus.png" class="toggler"
|
||||
id="toggle-1" style="display: none" alt="-" /></td>
|
||||
<td>
|
||||
<a href="numerical.html#module-numerical"><code class="xref">numerical</code></a></td><td>
|
||||
<em></em></td></tr>
|
||||
<tr>
|
||||
<td><img src="_static/minus.png" class="toggler"
|
||||
id="toggle-2" style="display: none" alt="-" /></td>
|
||||
<td>
|
||||
<code class="xref">numerical</code></td><td>
|
||||
<em></em></td></tr>
|
||||
<tr class="cg-2">
|
||||
<td></td>
|
||||
<td>   
|
||||
<a href="numerical.html#module-numerical.fit"><code class="xref">numerical.fit</code></a></td><td>
|
||||
<em></em></td></tr>
|
||||
<tr class="cg-2">
|
||||
<td></td>
|
||||
<td>   
|
||||
<a href="numerical.html#module-numerical.integration"><code class="xref">numerical.integration</code></a></td><td>
|
||||
<em></em></td></tr>
|
||||
<tr class="cg-2">
|
||||
<td></td>
|
||||
<td>   
|
||||
<a href="numerical.html#module-numerical.ode"><code class="xref">numerical.ode</code></a></td><td>
|
||||
<em></em></td></tr>
|
||||
<tr class="cg-2">
|
||||
<td></td>
|
||||
<td>   
|
||||
<a href="numerical.html#module-numerical.ode_model"><code class="xref">numerical.ode_model</code></a></td><td>
|
||||
<em></em></td></tr>
|
||||
<tr class="pcap"><td></td><td> </td><td></td></tr>
|
||||
<tr class="cap" id="cap-o"><td></td><td>
|
||||
<strong>o</strong></td><td></td></tr>
|
||||
<tr>
|
||||
<td></td>
|
||||
<td>
|
||||
<a href="numerical.html#module-ode"><code class="xref">ode</code></a> <em>(*nix, Windows)</em></td><td>
|
||||
<em>Numerical solver.</em></td></tr>
|
||||
<tr>
|
||||
<td></td>
|
||||
<td>
|
||||
<a href="solver_model.html#module-solver_model"><code class="xref">solver_model</code></a> <em>(*nix, Windows)</em></td><td>
|
||||
<em>Mechanical models.</em></td></tr>
|
||||
<a href="numerical.html#module-ode_model"><code class="xref">ode_model</code></a> <em>(*nix, Windows)</em></td><td>
|
||||
<em>Models of ordinary differential equations.</em></td></tr>
|
||||
</table>
|
||||
|
||||
|
||||
|
||||
2
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289
docs/build/html/solver.html
vendored
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docs/build/html/solver.html
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@@ -1,289 +0,0 @@
|
||||
|
||||
<!DOCTYPE html>
|
||||
|
||||
<html xmlns="http://www.w3.org/1999/xhtml">
|
||||
<head>
|
||||
<meta charset="utf-8" />
|
||||
<title>solver module — pylib 2019.5.19 documentation</title>
|
||||
<link rel="stylesheet" href="_static/alabaster.css" type="text/css" />
|
||||
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
|
||||
<link rel="stylesheet" href="_static/custom.css" type="text/css" />
|
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<script type="text/javascript" id="documentation_options" data-url_root="./" src="_static/documentation_options.js"></script>
|
||||
<script type="text/javascript" src="_static/jquery.js"></script>
|
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<script type="text/javascript" src="_static/underscore.js"></script>
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||||
<script type="text/javascript" src="_static/doctools.js"></script>
|
||||
<script type="text/javascript" src="_static/language_data.js"></script>
|
||||
<script async="async" type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-AMS-MML_HTMLorMML"></script>
|
||||
<link rel="index" title="Index" href="genindex.html" />
|
||||
<link rel="search" title="Search" href="search.html" />
|
||||
|
||||
<link rel="stylesheet" href="_static/custom.css" type="text/css" />
|
||||
|
||||
|
||||
<meta name="viewport" content="width=device-width, initial-scale=0.9, maximum-scale=0.9" />
|
||||
|
||||
</head><body>
|
||||
|
||||
|
||||
<div class="document">
|
||||
<div class="documentwrapper">
|
||||
<div class="bodywrapper">
|
||||
|
||||
|
||||
<div class="body" role="main">
|
||||
|
||||
<div class="section" id="module-solver">
|
||||
<span id="solver-module"></span><h1>solver module<a class="headerlink" href="#module-solver" title="Permalink to this headline">¶</a></h1>
|
||||
<p>Numerical solver of ordinary differential equations.</p>
|
||||
<p>Solves the initial value problem for systems of first order ordinary differential
|
||||
equations.</p>
|
||||
<dl class="field-list simple">
|
||||
<dt class="field-odd">Date</dt>
|
||||
<dd class="field-odd"><p>2015-09-21</p>
|
||||
</dd>
|
||||
</dl>
|
||||
<span class="target" id="module-solver"></span><dl class="function">
|
||||
<dt id="solver.e1">
|
||||
<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="headerlink" href="#solver.e1" title="Permalink to this definition">¶</a></dt>
|
||||
<dd><p>Explicit first-order method /
|
||||
(standard, or forward) Euler method /
|
||||
Runge-Kutta 1st order method.</p>
|
||||
<p>de:
|
||||
Euler’sche Polygonzugverfahren / explizite Euler-Verfahren /
|
||||
Euler-Cauchy-Verfahren / Euler-vorwärts-Verfahren</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>
|
||||
<p>Approximate the solution of the initial value problem</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[\begin{split}\dot{x} &= f(t,x) \\
|
||||
x(t_0) &= x_0\end{split}\]</div>
|
||||
<p>Choose a value h for the size of every step and set</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[t_i = t_0 + i h ~,\quad i=1,2,\ldots,n\]</div>
|
||||
<p>One step <span class="math notranslate nohighlight">\(h\)</span> of the Euler method from <span class="math notranslate nohighlight">\(t_i\)</span> to <span class="math notranslate nohighlight">\(t_{i+1}\)</span> is</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[\begin{split}x_{i+1} &= x_i + (t_{i+1}-t_i) f(t_i, x_i) \\
|
||||
x_{i+1} &= x_i + h f(t_i, x_i) \\\end{split}\]</div>
|
||||
<p>Example 1:</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[\begin{split}m\ddot{u} + d\dot{u} + ku = f(t) \\
|
||||
\ddot{u} = m^{-1}(f(t) - d\dot{u} - ku) \\\end{split}\]</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">
|
||||
\[\begin{split}\dot{x}_1 &= x_2 \\
|
||||
\dot{x}_2 &= m^{-1}(f(t) - d x_2 - k x_1) \\\end{split}\]</div>
|
||||
<p>or</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[\begin{split}\dot{x} &= f(t,x) \\
|
||||
\begin{bmatrix} \dot{x}_1 \\ \dot{x}_1 \end{bmatrix} &=
|
||||
\begin{bmatrix} x_2 \\ m^{-1}(f(t) - d x_2 - k x_1) \end{bmatrix} \\
|
||||
&=
|
||||
\begin{bmatrix} 0 \\ m^{-1} f(t) \end{bmatrix} +
|
||||
\begin{bmatrix} 0 & 1 \\ -m^{-1} k & -m^{-1} d \end{bmatrix}
|
||||
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\end{split}\]</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="solver.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="headerlink" href="#solver.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="solver.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="headerlink" href="#solver.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="solver.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="headerlink" href="#solver.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="solver.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="headerlink" href="#solver.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="solver.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="headerlink" href="#solver.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>
|
||||
|
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|
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|
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docs/build/html/solver_model.html
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<div class="section" id="module-solver_model">
|
||||
<span id="solver-model-module"></span><h1>solver_model module<a class="headerlink" href="#module-solver_model" title="Permalink to this headline">¶</a></h1>
|
||||
<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-solver_model"></span><dl class="function">
|
||||
<dt id="solver_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="headerlink" href="#solver_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="solver_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="headerlink" href="#solver_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="solver_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="headerlink" href="#solver_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>
|
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|
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#
|
||||
html_theme = 'alabaster'
|
||||
pygments_style = 'solarized-dark'
|
||||
|
||||
# Add any paths that contain custom static files (such as style sheets) here,
|
||||
# relative to this directory. They are copied after the builtin static files,
|
||||
@@ -66,6 +68,8 @@ html_theme_options = {
|
||||
'base_bg': '#292b2e',
|
||||
'base_text': '#b2b2b2',
|
||||
'body_text': '#b2b2b2',
|
||||
'viewcode_target_bg': '#292b2e',
|
||||
'pre_bg': '#25272c',
|
||||
'code_text': '#b2b2b2',
|
||||
'code_hover': 'transparent',
|
||||
'sidebar_text': '#b2b2b2',
|
||||
@@ -76,6 +80,6 @@ html_theme_options = {
|
||||
'highlight_bg': '#444F65',
|
||||
'xref_bg': 'transparent',
|
||||
'xref_border': 'transparent',
|
||||
'seealso_bg': '#3c3c3c',
|
||||
'seealso_bg': '#25272c',
|
||||
'seealso_border': '#2C2C2C',
|
||||
}
|
||||
|
||||
@@ -1,7 +0,0 @@
|
||||
fit module
|
||||
==========
|
||||
|
||||
.. automodule:: fit
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
@@ -6,7 +6,5 @@ src
|
||||
|
||||
data
|
||||
date
|
||||
fit
|
||||
geometry
|
||||
solver
|
||||
solver_model
|
||||
numerical
|
||||
|
||||
46
docs/source/numerical.rst
Normal file
46
docs/source/numerical.rst
Normal file
@@ -0,0 +1,46 @@
|
||||
numerical package
|
||||
=================
|
||||
|
||||
Submodules
|
||||
----------
|
||||
|
||||
numerical.fit module
|
||||
--------------------
|
||||
|
||||
.. automodule:: numerical.fit
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
|
||||
numerical.integration module
|
||||
----------------------------
|
||||
|
||||
.. automodule:: numerical.integration
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
|
||||
numerical.ode module
|
||||
--------------------
|
||||
|
||||
.. automodule:: numerical.ode
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
|
||||
numerical.ode\_model module
|
||||
---------------------------
|
||||
|
||||
.. automodule:: numerical.ode_model
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
|
||||
|
||||
Module contents
|
||||
---------------
|
||||
|
||||
.. automodule:: numerical
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
@@ -1,7 +0,0 @@
|
||||
solver module
|
||||
=============
|
||||
|
||||
.. automodule:: solver
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
@@ -1,7 +0,0 @@
|
||||
solver\_model module
|
||||
====================
|
||||
|
||||
.. automodule:: solver_model
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
@@ -33,6 +33,7 @@ def data_read(file_name, x_column, y_column):
|
||||
#print(filds)
|
||||
x.append(float(fields[x_column]))
|
||||
y.append(float(fields[y_column]))
|
||||
file.close()
|
||||
return x, y
|
||||
|
||||
def data_load(file_name, verbose=False):
|
||||
|
||||
0
src/numerical/__init__.py
Normal file
0
src/numerical/__init__.py
Normal file
@@ -9,9 +9,8 @@
|
||||
.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
|
||||
"""
|
||||
from __future__ import print_function
|
||||
from pylab import array, argmax, subplot, plot, title, xlim, show, gradient, exp, sqrt, log, linspace
|
||||
from pylab import array, argmax, gradient, exp, sqrt, log, linspace
|
||||
from scipy.optimize import curve_fit
|
||||
from data import *
|
||||
|
||||
def gauss(x, *p):
|
||||
"""Gauss distribution function.
|
||||
@@ -92,33 +91,5 @@ def gauss_fit(x, y, e=None, x_fit=None, verbose=False):
|
||||
|
||||
return y_fit, popt, FWHM
|
||||
|
||||
def main():
|
||||
"""test function"""
|
||||
file_name = "../tests/test_fit.dat"
|
||||
x, y = data_read(file_name, 3, 2)
|
||||
|
||||
subplot(2,2,1)
|
||||
plot(x, y, '.-')
|
||||
|
||||
subplot(2,2,2)
|
||||
dx = x[1] - x[0]
|
||||
dydx = gradient(y, dx) # central differences
|
||||
x_fit = linspace(x[0], x[-1], 150 if len(x) < 50 else len(x)*3)
|
||||
y_fit, popt, FWHM = gauss_fit(list(reversed(x)), list(reversed(dydx)), x_fit=x_fit, verbose=True) # x must increase!
|
||||
plot(x, dydx, '.-')
|
||||
plot(x_fit, y_fit, 'r:')
|
||||
title('FWHM: %f' % FWHM)
|
||||
|
||||
subplot(2,2,3)
|
||||
plot(x, y, '.-')
|
||||
xlim(popt[1]-2*FWHM, popt[1]+2*FWHM)
|
||||
|
||||
subplot(2,2,4)
|
||||
plot(x, dydx, '.-')
|
||||
plot(x_fit, y_fit, 'r:')
|
||||
xlim(popt[1]-2*FWHM, popt[1]+2*FWHM)
|
||||
|
||||
show()
|
||||
|
||||
if __name__ == "__main__":
|
||||
main()
|
||||
True
|
||||
151
src/numerical/integration.py
Normal file
151
src/numerical/integration.py
Normal file
@@ -0,0 +1,151 @@
|
||||
# -*- coding: utf-8 -*-
|
||||
"""Numerical integration, numerical quadrature.
|
||||
|
||||
de: numerische Integration, numerische Quadratur.
|
||||
|
||||
:Date: 2015-10-15
|
||||
|
||||
.. module:: integration
|
||||
:platform: *nix, Windows
|
||||
:synopsis: Numerical integration.
|
||||
|
||||
.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
|
||||
"""
|
||||
|
||||
from __future__ import division
|
||||
from numpy import linspace, trapz, zeros
|
||||
|
||||
def trapez(f, a=0, b=1, N=10, x=None, verbose=False,
|
||||
save_values=False):
|
||||
r"""
|
||||
Integration of :math:`f(x)` using the trapezoidal rule
|
||||
(Simpson's rule, Kepler's rule).
|
||||
|
||||
de: Trapezregel, Simpsonregel (Thomas Simpson), Keplersche
|
||||
Fassregel (Johannes Kepler)
|
||||
|
||||
:param f: function to integrate.
|
||||
:type f: function or list
|
||||
:param a: lower limit of integration (default = 0).
|
||||
:type a: float
|
||||
:param b: upper limit of integration (default = 1).
|
||||
:type b: float
|
||||
:param N: specify the number of subintervals.
|
||||
:type N: int
|
||||
:param x: variable of integration, necessary if f is a list
|
||||
(default = None).
|
||||
:type x: list
|
||||
:param verbose: print information (default = False)
|
||||
:type verbose: bool
|
||||
|
||||
:returns: the definite integral as approximated by trapezoidal
|
||||
rule.
|
||||
:rtype: float
|
||||
|
||||
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.
|
||||
|
||||
.. math::
|
||||
f_n(x) = f(a)+\frac{f(b)-f(a)}{b-a}(x-a)
|
||||
|
||||
.. math::
|
||||
I &= \int\limits_a^b f(x) \,\mathrm{d}x \\
|
||||
I &\approx \int\limits_a^b f_n(x) \,\mathrm{d}x \\
|
||||
&= \int\limits_a^b
|
||||
\left( f(a)+\frac{f(b)-f(a)}{b-a}(x-a) \right)
|
||||
\mathrm{d}x \\
|
||||
&= \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 \\
|
||||
&= \frac{b-a}{2}\left[f(a)+f(b)\right]
|
||||
|
||||
The composite trapezium rule. If the interval is divided into n
|
||||
segments (not necessarily equal)
|
||||
|
||||
.. math::
|
||||
a = x_0 \leq x_1 \leq x_2 \leq \ldots \leq x_n = b
|
||||
|
||||
.. math::
|
||||
I &\approx \sum\limits_{i=0}^{n-1} \frac{1}{2} (x_{i+1}-x_i)
|
||||
\left[f(x_{i+1})+f(x_i)\right] \\
|
||||
|
||||
Special Case (Equaliy spaced base points)
|
||||
|
||||
.. math::
|
||||
x_{i+1}-x_i = h \quad \forall i
|
||||
|
||||
.. math::
|
||||
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\}
|
||||
|
||||
.. rubric:: Example
|
||||
|
||||
.. math::
|
||||
I &= \int\limits_a^b f(x) \,\mathrm{d}x \\
|
||||
f(x) &= x^2 \\
|
||||
a &= 0 \\
|
||||
b &= 1
|
||||
|
||||
analytical solution
|
||||
|
||||
.. math::
|
||||
I = \int\limits_{0}^{1} x^2 \,\mathrm{d}x
|
||||
= \left. \frac{1}{3} x^3 \right\vert_0^1
|
||||
= \frac{1}{3}
|
||||
|
||||
numerical solution
|
||||
|
||||
>>> f = lambda(x): x**2
|
||||
>>> trapez(f, 0, 1, 1)
|
||||
0.5
|
||||
>>> trapez(f, 0, 1, 10)
|
||||
0.3350000000000001
|
||||
>>> trapez(f, 0, 1, 100)
|
||||
0.33335000000000004
|
||||
"""
|
||||
N = int(N)
|
||||
# f is function or list
|
||||
if hasattr(f, '__call__'):
|
||||
# h width of each subinterval
|
||||
h = (b-a)/N
|
||||
|
||||
# x variable of integration
|
||||
x = linspace(a, b, N+1)
|
||||
if save_values:
|
||||
# ff contribution from the points
|
||||
ff = zeros((N+1))
|
||||
for n in linspace(0, N, N+1):
|
||||
ff[n] = f(x[n])
|
||||
T = (ff[0]/2.+sum(ff[1:N])+ff[N]/2.)*h
|
||||
else:
|
||||
TL = f(x[0])
|
||||
TR = f(x[N])
|
||||
TI = 0
|
||||
for n in range(1, N):
|
||||
TI = TI + f(x[n])
|
||||
T = (TL/2.+TI+TR/2.)*h
|
||||
else:
|
||||
N = len(f)-1
|
||||
T = 0
|
||||
for n in range(N):
|
||||
T = T + (x[n+1]-x[n])/2*(f[n+1]+f[n])
|
||||
|
||||
if verbose:
|
||||
print(T)
|
||||
|
||||
return T
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
func = lambda x: x**2
|
||||
trapez(func, 0, 1, 1e6, verbose=True)
|
||||
#print(trapz(func, linspace(0,1,10)))
|
||||
|
||||
trapez([0,1,4,9], x=[0,1,2,3], verbose=True)
|
||||
#print(trapz([0,1,4,9]))
|
||||
|
||||
trapez([2,2], x=[-1,1], verbose=True)
|
||||
|
||||
trapez([-1,1,1,-1], x=[-1,-1,1,1], verbose=True)
|
||||
@@ -1,357 +1,424 @@
|
||||
#!/usr/bin/env python
|
||||
# -*- coding: utf-8 -*-
|
||||
"""Numerical solver of ordinary differential equations.
|
||||
|
||||
Solves the initial value problem for systems of first order ordinary differential
|
||||
equations.
|
||||
|
||||
:Date: 2015-09-21
|
||||
|
||||
.. module:: solver
|
||||
:platform: *nix, Windows
|
||||
:synopsis: Numerical solver.
|
||||
|
||||
.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
|
||||
"""
|
||||
|
||||
from __future__ import division, print_function
|
||||
from numpy import array, isnan, sum, zeros, dot
|
||||
from numpy.linalg import norm, inv
|
||||
|
||||
def e1(f, x0, t, *p, verbose=False):
|
||||
r"""Explicit first-order method /
|
||||
(standard, or forward) Euler method /
|
||||
Runge-Kutta 1st order method.
|
||||
|
||||
de:
|
||||
Euler'sche Polygonzugverfahren / explizite Euler-Verfahren /
|
||||
Euler-Cauchy-Verfahren / Euler-vorwärts-Verfahren
|
||||
|
||||
:param f: the function to solve
|
||||
:type f: function
|
||||
:param x0: initial condition
|
||||
:type x0: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function (thickness, diameter, ...)
|
||||
:param verbose: print information (default = False)
|
||||
:type verbose: bool
|
||||
|
||||
Approximate the solution of the initial value problem
|
||||
|
||||
.. math ::
|
||||
\dot{x} &= f(t,x) \\
|
||||
x(t_0) &= x_0
|
||||
|
||||
Choose a value h for the size of every step and set
|
||||
|
||||
.. math ::
|
||||
t_i = t_0 + i h ~,\quad i=1,2,\ldots,n
|
||||
|
||||
One step :math:`h` of the Euler method from :math:`t_i` to :math:`t_{i+1}` is
|
||||
|
||||
.. math ::
|
||||
x_{i+1} &= x_i + (t_{i+1}-t_i) f(t_i, x_i) \\
|
||||
x_{i+1} &= x_i + h f(t_i, x_i) \\
|
||||
|
||||
Example 1:
|
||||
|
||||
.. math ::
|
||||
m\ddot{u} + d\dot{u} + ku = f(t) \\
|
||||
\ddot{u} = m^{-1}(f(t) - d\dot{u} - ku) \\
|
||||
|
||||
with
|
||||
|
||||
.. math ::
|
||||
x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\
|
||||
x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\
|
||||
|
||||
becomes
|
||||
|
||||
.. math ::
|
||||
\dot{x}_1 &= x_2 \\
|
||||
\dot{x}_2 &= m^{-1}(f(t) - d x_2 - k x_1) \\
|
||||
|
||||
or
|
||||
|
||||
.. math ::
|
||||
\dot{x} &= f(t,x) \\
|
||||
\begin{bmatrix} \dot{x}_1 \\ \dot{x}_1 \end{bmatrix} &=
|
||||
\begin{bmatrix} x_2 \\ m^{-1}(f(t) - d x_2 - k x_1) \end{bmatrix} \\
|
||||
&=
|
||||
\begin{bmatrix} 0 \\ m^{-1} f(t) \end{bmatrix} +
|
||||
\begin{bmatrix} 0 & 1 \\ -m^{-1} k & -m^{-1} d \end{bmatrix}
|
||||
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}
|
||||
|
||||
Example 2:
|
||||
|
||||
.. math ::
|
||||
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) \\
|
||||
|
||||
with
|
||||
|
||||
.. math ::
|
||||
x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\
|
||||
x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\
|
||||
|
||||
becomes
|
||||
|
||||
.. math ::
|
||||
\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) \\
|
||||
|
||||
or
|
||||
|
||||
.. math ::
|
||||
\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}
|
||||
|
||||
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.
|
||||
"""
|
||||
x = zeros((len(t), len(x0))) # Preallocate array
|
||||
x[0,:] = x0 # Initial condition gives solution at t=0.
|
||||
for i in range(len(t)-1):
|
||||
Dt = t[i+1]-t[i]
|
||||
dxdt = array(f(x[i,:], t[i], *p))
|
||||
x[i+1,:] = x[i,:] + dxdt*Dt # Approximate solution at next value of x
|
||||
if verbose:
|
||||
print('Numerical integration of ODE using explicit first-order method (Euler / Runge-Kutta) was successful.')
|
||||
return x
|
||||
|
||||
def e2(f, x0, t, *p, verbose=False):
|
||||
r"""Explicit second-order method / Runge-Kutta 2nd order method.
|
||||
|
||||
:param f: the function to solve
|
||||
:type f: function
|
||||
:param x0: initial condition
|
||||
:type x0: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function (thickness, diameter, ...)
|
||||
:param verbose: print information (default = False)
|
||||
:type verbose: bool
|
||||
"""
|
||||
x = zeros((len(t), len(x0)))
|
||||
x[0,:] = x0 # initial condition
|
||||
for i in range(len(t)-1): # calculation loop
|
||||
Dt = t[i+1]-t[i]
|
||||
k_1 = array(f(x[i,:], t[i], *p))
|
||||
k_2 = array(f(x[i,:]+0.5*Dt*k_1, t[i]+0.5*Dt, *p))
|
||||
x[i+1,:] = x[i,:] + k_2*Dt # main equation
|
||||
if verbose:
|
||||
print('Numerical integration of ODE using explicit 2th-order method (Runge-Kutta) was successful.')
|
||||
return x
|
||||
|
||||
def e4(f, x0, t, *p, verbose=False):
|
||||
r"""Explicit fourth-order method / Runge-Kutta 4th order method.
|
||||
|
||||
:param f: the function to solve
|
||||
:type f: function
|
||||
:param x0: initial condition
|
||||
:type x0: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function (thickness, diameter, ...)
|
||||
:param verbose: print information (default = False)
|
||||
:type verbose: bool
|
||||
"""
|
||||
x = zeros((len(t), len(x0))) # Preallocate array
|
||||
x[0,:] = x0 # Initial condition gives solution at t=0.
|
||||
for i in range(len(t)-1): # calculation loop
|
||||
Dt = t[i+1]-t[i]
|
||||
k_1 = array(f(x[i,:], t[i], *p))
|
||||
k_2 = array(f(x[i,:]+0.5*Dt*k_1, t[i]+0.5*Dt, *p))
|
||||
k_3 = array(f(x[i,:]+0.5*Dt*k_2, t[i]+0.5*Dt, *p))
|
||||
k_4 = array(f(x[i,:]+k_3*Dt, t[i]+Dt, *p))
|
||||
x[i+1,:] = x[i,:] + 1./6*(k_1+2*k_2+2*k_3+k_4)*Dt # main equation
|
||||
if verbose:
|
||||
print('Numerical integration of ODE using explicit 4th-order method (Runge-Kutta) was successful.')
|
||||
return x
|
||||
|
||||
def i1(f, x0, t, *p, max_iterations=1000, tol=1e-9, verbose=False):
|
||||
r"""Implicite first-order method / backward Euler method.
|
||||
|
||||
:param f: the function to solve
|
||||
:type f: function
|
||||
:param x0: initial condition
|
||||
:type x0: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function (thickness, diameter, ...)
|
||||
:param max_iterations: maximum number of iterations
|
||||
:type max_iterations: int
|
||||
:param tol: tolerance against residuum (default = 1e-9)
|
||||
:type tol: float
|
||||
:param verbose: print information (default = False)
|
||||
:type verbose: bool
|
||||
|
||||
The backward Euler method has order one and is A-stable.
|
||||
"""
|
||||
iterations = zeros((len(t), 1))
|
||||
x = zeros((len(t), len(x0))) # Preallocate array
|
||||
x[0,:] = x0 # Initial condition gives solution at t=0.
|
||||
for i in range(len(t)-1):
|
||||
Dt = t[i+1]-t[i]
|
||||
x1 = x[i,:]
|
||||
for j in range(max_iterations): # fixed-point iteration
|
||||
dxdt = array(f(x1, t[i+1], *p))
|
||||
x11 = x[i,:] + dxdt*Dt # Approximate solution at next value of x
|
||||
residuum = norm(x11-x1)/norm(x11)
|
||||
x1 = x11
|
||||
if residuum < tol:
|
||||
break
|
||||
iterations[i] = j+1
|
||||
x[i+1,:] = x1
|
||||
if verbose:
|
||||
print('Numerical integration of ODE using implicite first-order method (Euler) was successful.')
|
||||
return x, iterations
|
||||
|
||||
def newmark_newtonraphson(f, x0, xp0, xpp0, t, *p, gamma=.5, beta=.25, max_iterations=1000, tol=1e-9, verbose=False):
|
||||
r"""Newmark method.
|
||||
|
||||
:param f: the function to solve
|
||||
:type f: function
|
||||
:param x0: initial condition
|
||||
:type x0: list
|
||||
:param xp0: initial condition
|
||||
:type xp0: list
|
||||
:param xpp0: initial condition
|
||||
:type xpp0: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function (thickness, diameter, ...)
|
||||
:param gamma: newmark parameter for velocity (default = 0.5)
|
||||
:type gamma: float
|
||||
:param beta: newmark parameter for displacement (default = 0.25)
|
||||
:type beta: float
|
||||
:param max_iterations: maximum number of iterations
|
||||
:type max_iterations: int
|
||||
:param tol: tolerance against residuum (default = 1e-9)
|
||||
:type tol: float
|
||||
:param verbose: print information (default = False)
|
||||
:type verbose: bool
|
||||
"""
|
||||
iterations = zeros((len(t), 1))
|
||||
x = zeros((len(t), len(x0))) # Preallocate array
|
||||
xp = zeros((len(t), len(xp0))) # Preallocate array
|
||||
xpp = zeros((len(t), len(xpp0))) # Preallocate array
|
||||
x[0,:] = x0 # Initial condition gives solution at t=0.
|
||||
xp[0,:] = xp0 # Initial condition gives solution at t=0.
|
||||
xpp[0,:] = xpp0 # Initial condition gives solution at t=0.
|
||||
for i in range(len(t)-1):
|
||||
Dt = t[i+1]-t[i]
|
||||
|
||||
xi = x[i,:].reshape(3,1)
|
||||
xpi = xp[i,:].reshape(3,1)
|
||||
xppi = xpp[i,:].reshape(3,1)
|
||||
x1 = xi
|
||||
xp1 = xpi
|
||||
xpp1 = xppi
|
||||
j = 0
|
||||
for j in range(max_iterations): # fixed-point iteration
|
||||
#dxdt = array(f(t[i+1], x1, p))
|
||||
#x11 = x[i,:] + dxdt*Dt # Approximate solution at next value of x
|
||||
|
||||
N, dN, dNp, dNpp = f(x1.reshape(-1,).tolist(),
|
||||
xp1.reshape(-1,).tolist(), xpp1.reshape(-1,).tolist(),
|
||||
t[i], *p)
|
||||
if isnan(sum(dN)) or isnan(sum(dNp)) or isnan(sum(dNpp)):
|
||||
print('divergiert')
|
||||
break
|
||||
|
||||
xpp11 = xpp1 - dot(inv(dNpp), (N + dot(dN, (x1-xi)) + dot(dNp, (xp1-xpi))))
|
||||
xp1 = xpi + Dt*( (1-gamma)*xppi + gamma*xpp11 )
|
||||
x1 = xi + Dt*xpi + Dt**2*( (.5-beta)*xppi + beta*xpp11 )
|
||||
|
||||
residuum = norm(xpp11-xpp1)/norm(xpp11)
|
||||
xpp1 = xpp11
|
||||
if residuum < tol:
|
||||
break
|
||||
iterations[i] = j+1
|
||||
|
||||
xpp[i+1,:] = xpp1.reshape(-1,).tolist()
|
||||
xp[i+1,:] = xp1.reshape(-1,).tolist()
|
||||
x[i+1,:] = x1.reshape(-1,).tolist()
|
||||
if verbose:
|
||||
print('Numerical integration of ODE using explicite newmark method was successful.')
|
||||
return x, xp, xpp, iterations
|
||||
# x = concatenate((x, xp, xpp), axis=1)
|
||||
|
||||
def newmark_newtonraphson_rdk(fnm, x0, xp0, xpp0, t, *p, gamma=.5, beta=.25, maxIterations=1000, tol=1e-9, verbose=False):
|
||||
r"""Newmark method.
|
||||
|
||||
:param f: the function to solve
|
||||
:type f: function
|
||||
:param x0: initial condition
|
||||
:type x0: list
|
||||
:param xp0: initial condition
|
||||
:type xp0: list
|
||||
:param xpp0: initial condition
|
||||
:type xpp0: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function (thickness, diameter, ...)
|
||||
:param gamma: newmark parameter for velocity (default = 0.5)
|
||||
:type gamma: float
|
||||
:param beta: newmark parameter for displacement (default = 0.25)
|
||||
:type beta: float
|
||||
:param max_iterations: maximum number of iterations
|
||||
:type max_iterations: int
|
||||
:param tol: tolerance against residuum (default = 1e-9)
|
||||
:type tol: float
|
||||
:param verbose: print information (default = False)
|
||||
:type verbose: bool
|
||||
"""
|
||||
iterations = zeros((len(t), 1))
|
||||
x = zeros((len(t), len(x0))) # Preallocate array
|
||||
xp = zeros((len(t), len(xp0))) # Preallocate array
|
||||
xpp = zeros((len(t), len(xpp0))) # Preallocate array
|
||||
x[0,:] = x0 # Initial condition gives solution at t=0.
|
||||
xp[0,:] = xp0 # Initial condition gives solution at t=0.
|
||||
xpp[0,:] = xpp0 # Initial condition gives solution at t=0.
|
||||
for i in range(len(t)-1):
|
||||
Dt = t[i+1]-t[i]
|
||||
|
||||
rm, rmx, rmxpp, rd, rdx, rdxp, rk, rkx, f = fnm(x[i,:], xp[i,:], xpp[i,:], t[i], *p)
|
||||
|
||||
xi = x[i,:].reshape(3,1)
|
||||
xpi = xp[i,:].reshape(3,1)
|
||||
xppi = xpp[i,:].reshape(3,1)
|
||||
x1 = xi
|
||||
xp1 = xpi
|
||||
xpp1 = xppi
|
||||
j = 0
|
||||
for j in range(maxIterations): # fixed-point iteration
|
||||
#dxdt = array(f(t[i+1], x1, p))
|
||||
#x11 = x[i,:] + dxdt*Dt # Approximate solution at next value of x
|
||||
|
||||
r = (rmx+rdx+rkx)*Dt**2./4 + rdxp*Dt/2 + rmxpp
|
||||
rp = f - (rm + dot(rmx, (Dt*xpi+Dt**2./4*xppi)) - dot(rmxpp, xppi) + \
|
||||
rd + dot(rdx, (Dt*xpi+Dt**2./4*xppi)) + dot(rdxp, Dt/2*xppi) + \
|
||||
rk + dot(rkx, (Dt*xpi+Dt**2./4*xppi)) )
|
||||
xpp11 = dot(inv(r), rp)
|
||||
xp1 = xpi + Dt*( (1-gamma)*xppi + gamma*xpp11 )
|
||||
x1 = xi + Dt*xpi + Dt**2*( (.5-beta)*xppi + beta*xpp11 )
|
||||
|
||||
residuum = norm(xpp11-xpp1)/norm(xpp11)
|
||||
xpp1 = xpp11
|
||||
if residuum < tol:
|
||||
break
|
||||
iterations[i] = j+1
|
||||
|
||||
xpp[i+1,:] = xpp1.reshape(-1,).tolist()
|
||||
xp[i+1,:] = xp1.reshape(-1,).tolist()
|
||||
x[i+1,:] = x1.reshape(-1,).tolist()
|
||||
if verbose:
|
||||
print('Numerical integration of ODE using explicite newmark method was successful.')
|
||||
return x, xp, xpp, iterations
|
||||
# x = concatenate((x, xp, xpp), axis=1)
|
||||
#!/usr/bin/env python
|
||||
# -*- coding: utf-8 -*-
|
||||
"""Numerical solver of ordinary differential equations.
|
||||
|
||||
Solves the initial value problem for systems of first order ordinary differential
|
||||
equations.
|
||||
|
||||
:Date: 2015-09-21
|
||||
|
||||
.. module:: ode
|
||||
:platform: *nix, Windows
|
||||
:synopsis: Numerical solver.
|
||||
|
||||
.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
|
||||
"""
|
||||
|
||||
from __future__ import division, print_function
|
||||
from numpy import array, isnan, sum, zeros, dot
|
||||
from numpy.linalg import norm, inv
|
||||
|
||||
def e1(f, x0, t, *p, verbose=False):
|
||||
r"""Explicit first-order method /
|
||||
(standard, or forward) Euler method /
|
||||
Runge-Kutta 1st order method.
|
||||
|
||||
de:
|
||||
Euler'sche Polygonzugverfahren / explizite Euler-Verfahren /
|
||||
Euler-Cauchy-Verfahren / Euler-vorwärts-Verfahren
|
||||
|
||||
:param f: the function to solve
|
||||
:type f: function
|
||||
:param x0: initial condition
|
||||
:type x0: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function (thickness, diameter, ...)
|
||||
:param verbose: print information (default = False)
|
||||
:type verbose: bool
|
||||
|
||||
Approximate the solution of the initial value problem
|
||||
|
||||
.. math ::
|
||||
\dot{x} &= f(t,x) \\
|
||||
x(t_0) &= x_0
|
||||
|
||||
Choose a value h for the size of every step and set
|
||||
|
||||
.. math ::
|
||||
t_i = t_0 + i h ~,\quad i=1,2,\ldots,n
|
||||
|
||||
The derivative of the solution is approximated as the forward difference
|
||||
equation
|
||||
|
||||
.. math ::
|
||||
\dot{x}_i = f(t_i, x_i) = \frac{x_{i+1} - x_i}{t_{i+1}-t_i}
|
||||
|
||||
Therefore one step :math:`h` of the Euler method from :math:`t_i` to
|
||||
:math:`t_{i+1}` is
|
||||
|
||||
.. math ::
|
||||
x_{i+1} &= x_i + (t_{i+1}-t_i) f(t_i, x_i) \\
|
||||
x_{i+1} &= x_i + h f(t_i, x_i) \\
|
||||
|
||||
Example 1:
|
||||
|
||||
.. math ::
|
||||
m\ddot{u} + d\dot{u} + ku = f(t) \\
|
||||
\ddot{u} = m^{-1}(f(t) - d\dot{u} - ku) \\
|
||||
|
||||
with
|
||||
|
||||
.. math ::
|
||||
x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\
|
||||
x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\
|
||||
|
||||
becomes
|
||||
|
||||
.. math ::
|
||||
\dot{x}_1 &= x_2 \\
|
||||
\dot{x}_2 &= m^{-1}(f(t) - d x_2 - k x_1) \\
|
||||
|
||||
or
|
||||
|
||||
.. math ::
|
||||
\dot{x} &= f(t,x) \\
|
||||
\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} \\
|
||||
&=
|
||||
\begin{bmatrix} 0 \\ m^{-1} f(t) \end{bmatrix} +
|
||||
\begin{bmatrix} 0 & 1 \\ -m^{-1} k & -m^{-1} d \end{bmatrix}
|
||||
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}
|
||||
|
||||
Example 2:
|
||||
|
||||
.. math ::
|
||||
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) \\
|
||||
|
||||
with
|
||||
|
||||
.. math ::
|
||||
x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\
|
||||
x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\
|
||||
|
||||
becomes
|
||||
|
||||
.. math ::
|
||||
\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) \\
|
||||
|
||||
or
|
||||
|
||||
.. math ::
|
||||
\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}
|
||||
|
||||
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.
|
||||
"""
|
||||
x = zeros((len(t), len(x0))) # Preallocate array
|
||||
x[0,:] = x0 # Initial condition gives solution at first t
|
||||
for i in range(len(t)-1): # Calculation loop
|
||||
Dt = t[i+1]-t[i]
|
||||
dxdt = array(f(x[i,:], t[i], *p))
|
||||
x[i+1,:] = x[i,:] + dxdt*Dt # Approximate solution at next value of x
|
||||
if verbose:
|
||||
print('Numerical integration of ODE using explicit first-order method (Euler / Runge-Kutta) was successful.')
|
||||
return x
|
||||
|
||||
def e2(f, x0, t, *p, verbose=False):
|
||||
r"""Explicit second-order method / Runge-Kutta 2nd order method.
|
||||
|
||||
:param f: the function to solve
|
||||
:type f: function
|
||||
:param x0: initial condition
|
||||
:type x0: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function (thickness, diameter, ...)
|
||||
:param verbose: print information (default = False)
|
||||
:type verbose: bool
|
||||
"""
|
||||
x = zeros((len(t), len(x0))) # Preallocate array
|
||||
x[0,:] = x0 # Initial condition gives solution at first t
|
||||
for i in range(len(t)-1): # Calculation loop
|
||||
Dt = t[i+1]-t[i]
|
||||
k_1 = array(f(x[i,:], t[i], *p))
|
||||
k_2 = array(f(x[i,:]+0.5*Dt*k_1, t[i]+0.5*Dt, *p))
|
||||
x[i+1,:] = x[i,:] + k_2*Dt # Approximate solution at next value of x
|
||||
if verbose:
|
||||
print('Numerical integration of ODE using explicit 2th-order method (Runge-Kutta) was successful.')
|
||||
return x
|
||||
|
||||
def e4(f, x0, t, *p, verbose=False):
|
||||
r"""Explicit fourth-order method / Runge-Kutta 4th order method.
|
||||
|
||||
:param f: the function to solve
|
||||
:type f: function
|
||||
:param x0: initial condition
|
||||
:type x0: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function (thickness, diameter, ...)
|
||||
:param verbose: print information (default = False)
|
||||
:type verbose: bool
|
||||
"""
|
||||
x = zeros((len(t), len(x0))) # Preallocate array
|
||||
x[0,:] = x0 # Initial condition
|
||||
for i in range(len(t)-1): # Calculation loop
|
||||
Dt = t[i+1]-t[i]
|
||||
k_1 = array(f(x[i,:], t[i], *p))
|
||||
k_2 = array(f(x[i,:]+0.5*Dt*k_1, t[i]+0.5*Dt, *p))
|
||||
k_3 = array(f(x[i,:]+0.5*Dt*k_2, t[i]+0.5*Dt, *p))
|
||||
k_4 = array(f(x[i,:]+k_3*Dt, t[i]+Dt, *p))
|
||||
x[i+1,:] = x[i,:] + 1./6*(k_1+2*k_2+2*k_3+k_4)*Dt # Approximate solution at next value of x
|
||||
if verbose:
|
||||
print('Numerical integration of ODE using explicit 4th-order method (Runge-Kutta) was successful.')
|
||||
return x
|
||||
|
||||
def dxdt_Dt(f, x, t, Dt, *p):
|
||||
r"""
|
||||
:param f: :math:`f = \dot{x}`
|
||||
:type f: function
|
||||
:param Dt: :math:`\Delta{t}`
|
||||
|
||||
:returns: :math:`\Delta x = \dot{x} \Delta t`
|
||||
"""
|
||||
return array(dxdt(x, t, *p)) * Dt
|
||||
|
||||
def fixed_point_iteration(f, xi, t, max_iterations=1000, tol=1e-9, verbose=False):
|
||||
r"""
|
||||
:param f: the function to iterate :math:`f = \Delta{x}(t)`
|
||||
:type f: function
|
||||
:param xi: initial condition :math:`x_i`
|
||||
:type xi: list
|
||||
:param t: time :math:`t`
|
||||
:type t: float
|
||||
:param `*p`: parameters of the function (thickness, diameter, ...)
|
||||
:param max_iterations: maximum number of iterations
|
||||
:type max_iterations: int
|
||||
:param tol: tolerance against residuum (default = 1e-9)
|
||||
:type tol: float
|
||||
:param verbose: print information (default = False)
|
||||
:type verbose: bool
|
||||
|
||||
:returns: :math:`x_{i+1}`
|
||||
|
||||
.. math ::
|
||||
x_{i+1} = x_i + \Delta x
|
||||
|
||||
.. seealso::
|
||||
:meth:`dxdt_Dt` for :math:`\Delta x`
|
||||
"""
|
||||
xi = x0
|
||||
for j in range(max_iterations): # Fixed-point iteration
|
||||
Dx = array(f(xi, t, *p))
|
||||
xi1 = x0 + Dx # Approximate solution at next value of x
|
||||
residuum = norm(xi1-xi)/norm(xi1)
|
||||
xi = xi1
|
||||
if residuum < tol:
|
||||
break
|
||||
iterations = j+1 # number beginning with 1 therefore + 1
|
||||
return xi, iterations
|
||||
|
||||
def i1n(f, x0, t, *p, max_iterations=1000, tol=1e-9, verbose=False):
|
||||
iterations = zeros((len(t), 1))
|
||||
x = zeros((len(t), len(x0))) # Preallocate array
|
||||
x[0,:] = x0 # Initial condition gives solution at first t
|
||||
for i in range(len(t)-1):
|
||||
Dt = t[i+1]-t[i]
|
||||
xi = x[i,:]
|
||||
Dx = dxdt_Dt(f, xi, t[i+1], Dt, *p)
|
||||
x[i+1,:], iterations[i] = fixed_point_iteration(Dx, xi, t, max_iterations, tol, verbose)
|
||||
if verbose:
|
||||
print('Numerical integration of ODE using implicite first-order method (Euler) was successful.')
|
||||
return x, iterations
|
||||
|
||||
def i1(f, x0, t, *p, max_iterations=1000, tol=1e-9, verbose=False):
|
||||
r"""Implicite first-order method / backward Euler method.
|
||||
|
||||
:param f: the function to solve
|
||||
:type f: function
|
||||
:param x0: initial condition
|
||||
:type x0: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function (thickness, diameter, ...)
|
||||
:param max_iterations: maximum number of iterations
|
||||
:type max_iterations: int
|
||||
:param tol: tolerance against residuum (default = 1e-9)
|
||||
:type tol: float
|
||||
:param verbose: print information (default = False)
|
||||
:type verbose: bool
|
||||
|
||||
The backward Euler method has order one and is A-stable.
|
||||
"""
|
||||
iterations = zeros((len(t), 1))
|
||||
x = zeros((len(t), len(x0))) # Preallocate array
|
||||
x[0,:] = x0 # Initial condition gives solution at first t
|
||||
for i in range(len(t)-1):
|
||||
Dt = t[i+1]-t[i]
|
||||
xi = x[i,:]
|
||||
# x(i+1) = x(i) + f(x(i+1), t(i+1)), exact value of f(x(i+1), t(i+1)) is not
|
||||
# available therefor using Newton-Raphson method
|
||||
for j in range(max_iterations): # Fixed-point iteration
|
||||
dxdt = array(f(xi, t[i+1], *p))
|
||||
xi1 = x[i,:] + dxdt*Dt # Approximate solution at next value of x
|
||||
residuum = norm(xi1-xi)/norm(xi1)
|
||||
xi = xi1
|
||||
if residuum < tol:
|
||||
break
|
||||
iterations[i] = j+1
|
||||
x[i+1,:] = xi
|
||||
if verbose:
|
||||
print('Numerical integration of ODE using implicite first-order method (Euler) was successful.')
|
||||
return x, iterations
|
||||
|
||||
def newmark_newtonraphson(f, x0, xp0, xpp0, t, *p, gamma=.5, beta=.25, max_iterations=1000, tol=1e-9, verbose=False):
|
||||
r"""Newmark method.
|
||||
|
||||
:param f: the function to solve
|
||||
:type f: function
|
||||
:param x0: initial condition
|
||||
:type x0: list
|
||||
:param xp0: initial condition
|
||||
:type xp0: list
|
||||
:param xpp0: initial condition
|
||||
:type xpp0: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function (thickness, diameter, ...)
|
||||
:param gamma: newmark parameter for velocity (default = 0.5)
|
||||
:type gamma: float
|
||||
:param beta: newmark parameter for displacement (default = 0.25)
|
||||
:type beta: float
|
||||
:param max_iterations: maximum number of iterations
|
||||
:type max_iterations: int
|
||||
:param tol: tolerance against residuum (default = 1e-9)
|
||||
:type tol: float
|
||||
:param verbose: print information (default = False)
|
||||
:type verbose: bool
|
||||
"""
|
||||
iterations = zeros((len(t), 1))
|
||||
x = zeros((len(t), len(x0))) # Preallocate array
|
||||
xp = zeros((len(t), len(xp0))) # Preallocate array
|
||||
xpp = zeros((len(t), len(xpp0))) # Preallocate array
|
||||
x[0,:] = x0 # Initial condition gives solution at first t
|
||||
xp[0,:] = xp0 # Initial condition gives solution at first t
|
||||
xpp[0,:] = xpp0 # Initial condition gives solution at first t
|
||||
for i in range(len(t)-1):
|
||||
Dt = t[i+1]-t[i]
|
||||
|
||||
xi = x[i,:].reshape(3,1)
|
||||
xpi = xp[i,:].reshape(3,1)
|
||||
xppi = xpp[i,:].reshape(3,1)
|
||||
x1 = xi
|
||||
xp1 = xpi
|
||||
xpp1 = xppi
|
||||
j = 0
|
||||
for j in range(max_iterations): # Fixed-point iteration
|
||||
#dxdt = array(f(t[i+1], x1, p))
|
||||
#x11 = x[i,:] + dxdt*Dt # Approximate solution at next value of x
|
||||
|
||||
N, dN, dNp, dNpp = f(x1.reshape(-1,).tolist(),
|
||||
xp1.reshape(-1,).tolist(), xpp1.reshape(-1,).tolist(),
|
||||
t[i], *p)
|
||||
if isnan(sum(dN)) or isnan(sum(dNp)) or isnan(sum(dNpp)):
|
||||
print('divergiert')
|
||||
break
|
||||
|
||||
xpp11 = xpp1 - dot(inv(dNpp), (N + dot(dN, (x1-xi)) + dot(dNp, (xp1-xpi))))
|
||||
xp1 = xpi + Dt*( (1-gamma)*xppi + gamma*xpp11 )
|
||||
x1 = xi + Dt*xpi + Dt**2*( (.5-beta)*xppi + beta*xpp11 )
|
||||
|
||||
residuum = norm(xpp11-xpp1)/norm(xpp11)
|
||||
xpp1 = xpp11
|
||||
if residuum < tol:
|
||||
break
|
||||
iterations[i] = j+1
|
||||
|
||||
xpp[i+1,:] = xpp1.reshape(-1,).tolist()
|
||||
xp[i+1,:] = xp1.reshape(-1,).tolist()
|
||||
x[i+1,:] = x1.reshape(-1,).tolist()
|
||||
if verbose:
|
||||
print('Numerical integration of ODE using explicite newmark method was successful.')
|
||||
return x, xp, xpp, iterations
|
||||
# x = concatenate((x, xp, xpp), axis=1)
|
||||
|
||||
def newmark_newtonraphson_rdk(fnm, x0, xp0, xpp0, t, *p, gamma=.5, beta=.25, maxIterations=1000, tol=1e-9, verbose=False):
|
||||
r"""Newmark method.
|
||||
|
||||
:param f: the function to solve
|
||||
:type f: function
|
||||
:param x0: initial condition
|
||||
:type x0: list
|
||||
:param xp0: initial condition
|
||||
:type xp0: list
|
||||
:param xpp0: initial condition
|
||||
:type xpp0: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function (thickness, diameter, ...)
|
||||
:param gamma: newmark parameter for velocity (default = 0.5)
|
||||
:type gamma: float
|
||||
:param beta: newmark parameter for displacement (default = 0.25)
|
||||
:type beta: float
|
||||
:param max_iterations: maximum number of iterations
|
||||
:type max_iterations: int
|
||||
:param tol: tolerance against residuum (default = 1e-9)
|
||||
:type tol: float
|
||||
:param verbose: print information (default = False)
|
||||
:type verbose: bool
|
||||
"""
|
||||
iterations = zeros((len(t), 1))
|
||||
x = zeros((len(t), len(x0))) # Preallocate array
|
||||
xp = zeros((len(t), len(xp0))) # Preallocate array
|
||||
xpp = zeros((len(t), len(xpp0))) # Preallocate array
|
||||
x[0,:] = x0 # Initial condition gives solution at first t
|
||||
xp[0,:] = xp0 # Initial condition gives solution at first t
|
||||
xpp[0,:] = xpp0 # Initial condition gives solution at first t
|
||||
for i in range(len(t)-1):
|
||||
Dt = t[i+1]-t[i]
|
||||
|
||||
rm, rmx, rmxpp, rd, rdx, rdxp, rk, rkx, f = fnm(x[i,:], xp[i,:], xpp[i,:], t[i], *p)
|
||||
|
||||
xi = x[i,:].reshape(3,1)
|
||||
xpi = xp[i,:].reshape(3,1)
|
||||
xppi = xpp[i,:].reshape(3,1)
|
||||
x1 = xi
|
||||
xp1 = xpi
|
||||
xpp1 = xppi
|
||||
j = 0
|
||||
for j in range(maxIterations): # Fixed-point iteration
|
||||
#dxdt = array(f(t[i+1], x1, p))
|
||||
#x11 = x[i,:] + dxdt*Dt # Approximate solution at next value of x
|
||||
|
||||
r = (rmx+rdx+rkx)*Dt**2./4 + rdxp*Dt/2 + rmxpp
|
||||
rp = f - (rm + dot(rmx, (Dt*xpi+Dt**2./4*xppi)) - dot(rmxpp, xppi) + \
|
||||
rd + dot(rdx, (Dt*xpi+Dt**2./4*xppi)) + dot(rdxp, Dt/2*xppi) + \
|
||||
rk + dot(rkx, (Dt*xpi+Dt**2./4*xppi)) )
|
||||
xpp11 = dot(inv(r), rp)
|
||||
xp1 = xpi + Dt*( (1-gamma)*xppi + gamma*xpp11 )
|
||||
x1 = xi + Dt*xpi + Dt**2*( (.5-beta)*xppi + beta*xpp11 )
|
||||
|
||||
residuum = norm(xpp11-xpp1)/norm(xpp11)
|
||||
xpp1 = xpp11
|
||||
if residuum < tol:
|
||||
break
|
||||
iterations[i] = j+1
|
||||
|
||||
xpp[i+1,:] = xpp1.reshape(-1,).tolist()
|
||||
xp[i+1,:] = xp1.reshape(-1,).tolist()
|
||||
x[i+1,:] = x1.reshape(-1,).tolist()
|
||||
if verbose:
|
||||
print('Numerical integration of ODE using explicite newmark method was successful.')
|
||||
return x, xp, xpp, iterations
|
||||
# x = concatenate((x, xp, xpp), axis=1)
|
||||
@@ -1,120 +1,120 @@
|
||||
#!/usr/bin/env python
|
||||
# -*- coding: utf-8 -*-
|
||||
"""Mathmatical models governed by ordinary differential equations.
|
||||
|
||||
Describes initial value problems as systems of first order ordinary differential
|
||||
equations.
|
||||
|
||||
:Date: 2019-05-25
|
||||
|
||||
.. module:: solver_model
|
||||
:platform: *nix, Windows
|
||||
:synopsis: Mechanical models.
|
||||
|
||||
.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
|
||||
"""
|
||||
from __future__ import division, print_function
|
||||
from numpy import array, cos, sin, dot, square
|
||||
from numpy.linalg import inv
|
||||
|
||||
def disk(x, t, *p):
|
||||
"""Rotation of an eccentric disk.
|
||||
|
||||
:param x: values of the function
|
||||
:type x: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function
|
||||
|
||||
* diameter
|
||||
* eccentricity
|
||||
* torque
|
||||
"""
|
||||
qp1 = x[3]
|
||||
qp2 = x[4]
|
||||
qp3 = x[5]
|
||||
M = array([[1, 0, cos(x[2])], [0, 1, -sin(x[2])], [0, 0, 1]])
|
||||
y = array([[-2*p[0]*x[3]+sin(x[2])*x[5]**2-2*p[0]*cos(x[2])*x[5]-x[0]], \
|
||||
[-2*p[0]*x[4]+cos(x[2])*x[5]**2-2*p[0]*sin(x[2])*x[5]-x[1]], \
|
||||
[p[2]-p[1]*x[1]*sin(x[2])+p[1]*x[0]*cos(x[2])]])
|
||||
qp46 = dot(inv(M), y)
|
||||
qp4, qp5, qp6 = qp46.reshape(-1,).tolist() # 2d array to 1d array to list
|
||||
return qp1, qp2, qp3, qp4, qp5, qp6
|
||||
|
||||
def disk_nm(xn, xpn, xppn, t, *p):
|
||||
"""Rotation of an eccentric disk.
|
||||
|
||||
:param xn: values of the function
|
||||
:type xn: list
|
||||
:param xpn: first derivative values of the function
|
||||
:type xpn: list
|
||||
:param xppn: second derivative values of the function
|
||||
:type xppn: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function
|
||||
|
||||
* diameter
|
||||
* eccentricity
|
||||
* torque
|
||||
"""
|
||||
N = array([[xppn[0]+cos(xn[2])*xppn[2]+2*p[0]*xpn[0]+2*p[0]*cos(xn[2])*xpn[2]-sin(xn[2])*square(xpn[2])+xn[0]],
|
||||
[xppn[1]-sin(xn[2])*xppn[2]+2*p[0]*xpn[1]-2*p[0]*sin(xn[2])*xpn[2]-cos(xn[2])*square(xpn[2])+xn[1]],
|
||||
[xppn[2]+p[1]*(-cos(xn[2])*xn[0]+sin(xn[2])*xn[1])-p[2]]])
|
||||
dN = array([[1, 0, -sin(xn[2]*xppn[2])-2*p[0]*sin(xn[2])*xpn[2]-cos(xn[2])*square(xpn[2])],
|
||||
[0, 1, -cos(xn[2]*xppn[2])-2*p[0]*cos(xn[2])*xpn[2]+sin(xn[2])*square(xpn[2])],
|
||||
[-p[1]*cos(xn[2]), p[1]*cos(xn[2]), p[1]*(sin(xn[2])*xn[0]+cos(xn[2])*xn[1])]])
|
||||
dNp = array([[2*p[0], 0, 2*p[0]*cos(xn[2])-2*sin(xn[2])*xpn[2]],
|
||||
[0, 2*p[0], -2*p[0]*sin(xn[2])-2*cos(xn[2])*xpn[2]],
|
||||
[0, 0, 0]])
|
||||
dNpp = array([[1, 0, cos(xn[2])],
|
||||
[0, 1, -sin(xn[2])],
|
||||
[0, 0, 1]])
|
||||
return N, dN, dNp, dNpp
|
||||
|
||||
def disk_nmmdk(xn, xpn, xppn, t, *p):
|
||||
"""Rotation of an eccentric disk.
|
||||
|
||||
:param xn: values of the function
|
||||
:type xn: list
|
||||
:param xpn: derivative values of the function
|
||||
:type xpn: list
|
||||
:param xppn: second derivative values of the function
|
||||
:type xppn: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function
|
||||
|
||||
* diameter
|
||||
* eccentricity
|
||||
* torque
|
||||
"""
|
||||
rm = array([[xppn[0]+cos(xn[2])*xppn[2]],
|
||||
[xppn[1]-sin(xn[2])*xppn[2]],
|
||||
[xppn[2]]])
|
||||
rmx = array([[0, 0, -sin(xn[2]*xppn[2])],
|
||||
[0, 0, -cos(xn[2]*xppn[2])],
|
||||
[0, 0, 0]])
|
||||
rmxpp = array([[1, 0, cos(xn[2])],
|
||||
[0, 1, -sin(xn[2])],
|
||||
[0, 0, 1]])
|
||||
rd = array([[2*p[0]*xpn[0]+2*p[0]*cos(xn[2])*xpn[2]-sin(xn[2])*square(xpn[2])],
|
||||
[2*p[0]*xpn[1]-2*p[0]*sin(xn[2])*xpn[2]-cos(xn[2])*square(xpn[2])],
|
||||
[0]])
|
||||
rdx = array([[0, 0, -2*p[0]*sin(xn[2])*xpn[2]-cos(xn[2])*square(xpn[2])],
|
||||
[0, 0, -2*p[0]*cos(xn[2])*xpn[2]+sin(xn[2])*square(xpn[2])],
|
||||
[0, 0, 0]])
|
||||
rdxp = array([[2*p[0], 0, 2*p[0]*cos(xn[2])-2*sin(xn[2])*xpn[2]],
|
||||
[0, 2*p[0], -2*p[0]*sin(xn[2])-2*cos(xn[2])*xpn[2]],
|
||||
[0, 0, 0]])
|
||||
rk = array([[xn[0]],
|
||||
[xn[1]],
|
||||
[p[1]*(-cos(xn[2])*xn[0]+sin(xn[2])*xn[1])]])
|
||||
rkx = array([[ 1, 0, 0],
|
||||
[ 0, 1, 0],
|
||||
[-p[1]*cos(xn[2]), p[1]*cos(xn[2]), p[1]*(sin(xn[2])*xn[0]+cos(xn[2])*xn[1])]])
|
||||
f = array([[0], [0], [p[2]]])
|
||||
return rm, rmx, rmxpp, rd, rdx, rdxp, rk, rkx, f
|
||||
|
||||
if __name__ == '__main__':
|
||||
0
|
||||
#!/usr/bin/env python
|
||||
# -*- coding: utf-8 -*-
|
||||
"""Mathmatical models governed by ordinary differential equations.
|
||||
|
||||
Describes initial value problems as systems of first order ordinary differential
|
||||
equations.
|
||||
|
||||
:Date: 2019-05-25
|
||||
|
||||
.. module:: ode_model
|
||||
:platform: *nix, Windows
|
||||
:synopsis: Models of ordinary differential equations.
|
||||
|
||||
.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
|
||||
"""
|
||||
from __future__ import division, print_function
|
||||
from numpy import array, cos, sin, dot, square
|
||||
from numpy.linalg import inv
|
||||
|
||||
def disk(x, t, *p):
|
||||
"""Rotation of an eccentric disk.
|
||||
|
||||
:param x: values of the function
|
||||
:type x: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function
|
||||
|
||||
* diameter
|
||||
* eccentricity
|
||||
* torque
|
||||
"""
|
||||
qp1 = x[3]
|
||||
qp2 = x[4]
|
||||
qp3 = x[5]
|
||||
M = array([[1, 0, cos(x[2])], [0, 1, -sin(x[2])], [0, 0, 1]])
|
||||
y = array([[-2*p[0]*x[3]+sin(x[2])*x[5]**2-2*p[0]*cos(x[2])*x[5]-x[0]], \
|
||||
[-2*p[0]*x[4]+cos(x[2])*x[5]**2-2*p[0]*sin(x[2])*x[5]-x[1]], \
|
||||
[p[2]-p[1]*x[1]*sin(x[2])+p[1]*x[0]*cos(x[2])]])
|
||||
qp46 = dot(inv(M), y)
|
||||
qp4, qp5, qp6 = qp46.reshape(-1,).tolist() # 2d array to 1d array to list
|
||||
return qp1, qp2, qp3, qp4, qp5, qp6
|
||||
|
||||
def disk_nm(xn, xpn, xppn, t, *p):
|
||||
"""Rotation of an eccentric disk.
|
||||
|
||||
:param xn: values of the function
|
||||
:type xn: list
|
||||
:param xpn: first derivative values of the function
|
||||
:type xpn: list
|
||||
:param xppn: second derivative values of the function
|
||||
:type xppn: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function
|
||||
|
||||
* diameter
|
||||
* eccentricity
|
||||
* torque
|
||||
"""
|
||||
N = array([[xppn[0]+cos(xn[2])*xppn[2]+2*p[0]*xpn[0]+2*p[0]*cos(xn[2])*xpn[2]-sin(xn[2])*square(xpn[2])+xn[0]],
|
||||
[xppn[1]-sin(xn[2])*xppn[2]+2*p[0]*xpn[1]-2*p[0]*sin(xn[2])*xpn[2]-cos(xn[2])*square(xpn[2])+xn[1]],
|
||||
[xppn[2]+p[1]*(-cos(xn[2])*xn[0]+sin(xn[2])*xn[1])-p[2]]])
|
||||
dN = array([[1, 0, -sin(xn[2]*xppn[2])-2*p[0]*sin(xn[2])*xpn[2]-cos(xn[2])*square(xpn[2])],
|
||||
[0, 1, -cos(xn[2]*xppn[2])-2*p[0]*cos(xn[2])*xpn[2]+sin(xn[2])*square(xpn[2])],
|
||||
[-p[1]*cos(xn[2]), p[1]*cos(xn[2]), p[1]*(sin(xn[2])*xn[0]+cos(xn[2])*xn[1])]])
|
||||
dNp = array([[2*p[0], 0, 2*p[0]*cos(xn[2])-2*sin(xn[2])*xpn[2]],
|
||||
[0, 2*p[0], -2*p[0]*sin(xn[2])-2*cos(xn[2])*xpn[2]],
|
||||
[0, 0, 0]])
|
||||
dNpp = array([[1, 0, cos(xn[2])],
|
||||
[0, 1, -sin(xn[2])],
|
||||
[0, 0, 1]])
|
||||
return N, dN, dNp, dNpp
|
||||
|
||||
def disk_nmmdk(xn, xpn, xppn, t, *p):
|
||||
"""Rotation of an eccentric disk.
|
||||
|
||||
:param xn: values of the function
|
||||
:type xn: list
|
||||
:param xpn: derivative values of the function
|
||||
:type xpn: list
|
||||
:param xppn: second derivative values of the function
|
||||
:type xppn: list
|
||||
:param t: time
|
||||
:type t: list
|
||||
:param `*p`: parameters of the function
|
||||
|
||||
* diameter
|
||||
* eccentricity
|
||||
* torque
|
||||
"""
|
||||
rm = array([[xppn[0]+cos(xn[2])*xppn[2]],
|
||||
[xppn[1]-sin(xn[2])*xppn[2]],
|
||||
[xppn[2]]])
|
||||
rmx = array([[0, 0, -sin(xn[2]*xppn[2])],
|
||||
[0, 0, -cos(xn[2]*xppn[2])],
|
||||
[0, 0, 0]])
|
||||
rmxpp = array([[1, 0, cos(xn[2])],
|
||||
[0, 1, -sin(xn[2])],
|
||||
[0, 0, 1]])
|
||||
rd = array([[2*p[0]*xpn[0]+2*p[0]*cos(xn[2])*xpn[2]-sin(xn[2])*square(xpn[2])],
|
||||
[2*p[0]*xpn[1]-2*p[0]*sin(xn[2])*xpn[2]-cos(xn[2])*square(xpn[2])],
|
||||
[0]])
|
||||
rdx = array([[0, 0, -2*p[0]*sin(xn[2])*xpn[2]-cos(xn[2])*square(xpn[2])],
|
||||
[0, 0, -2*p[0]*cos(xn[2])*xpn[2]+sin(xn[2])*square(xpn[2])],
|
||||
[0, 0, 0]])
|
||||
rdxp = array([[2*p[0], 0, 2*p[0]*cos(xn[2])-2*sin(xn[2])*xpn[2]],
|
||||
[0, 2*p[0], -2*p[0]*sin(xn[2])-2*cos(xn[2])*xpn[2]],
|
||||
[0, 0, 0]])
|
||||
rk = array([[xn[0]],
|
||||
[xn[1]],
|
||||
[p[1]*(-cos(xn[2])*xn[0]+sin(xn[2])*xn[1])]])
|
||||
rkx = array([[ 1, 0, 0],
|
||||
[ 0, 1, 0],
|
||||
[-p[1]*cos(xn[2]), p[1]*cos(xn[2]), p[1]*(sin(xn[2])*xn[0]+cos(xn[2])*xn[1])]])
|
||||
f = array([[0], [0], [p[2]]])
|
||||
return rm, rmx, rmxpp, rd, rdx, rdxp, rk, rkx, f
|
||||
|
||||
if __name__ == '__main__':
|
||||
True
|
||||
@@ -1,15 +1,53 @@
|
||||
"""Test of fit module.
|
||||
|
||||
:Date: 2019-05-29
|
||||
|
||||
.. module:: test_fit
|
||||
:platform: *nix, Windows
|
||||
:synopsis: Test of fit module.
|
||||
|
||||
.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
|
||||
"""
|
||||
import unittest
|
||||
|
||||
import os
|
||||
import sys
|
||||
from pylab import array, argmax, subplot, plot, title, xlim, show, gradient, linspace
|
||||
sys.path.insert(0, os.path.abspath('../src'))
|
||||
import fit
|
||||
from data import data_read
|
||||
from numerical.fit import gauss_fit
|
||||
|
||||
|
||||
class TestFit(unittest.TestCase):
|
||||
|
||||
def test_property(self):
|
||||
self.assertEqual()
|
||||
def test_gauss(self):
|
||||
"""test function"""
|
||||
file_name = "test_fit.dat"
|
||||
x, y = data_read(file_name, 3, 2)
|
||||
|
||||
subplot(2,2,1)
|
||||
plot(x, y, '.-')
|
||||
|
||||
subplot(2,2,2)
|
||||
dx = x[1] - x[0]
|
||||
dydx = gradient(y, dx) # central differences
|
||||
x_fit = linspace(x[0], x[-1], 150 if len(x) < 50 else len(x)*3)
|
||||
y_fit, popt, FWHM = gauss_fit(list(reversed(x)), list(reversed(dydx)), x_fit=x_fit, verbose=True) # x must increase!
|
||||
plot(x, dydx, '.-')
|
||||
plot(x_fit, y_fit, 'r:')
|
||||
title('FWHM: %f' % FWHM)
|
||||
|
||||
subplot(2,2,3)
|
||||
plot(x, y, '.-')
|
||||
xlim(popt[1]-2*FWHM, popt[1]+2*FWHM)
|
||||
|
||||
subplot(2,2,4)
|
||||
plot(x, dydx, '.-')
|
||||
plot(x_fit, y_fit, 'r:')
|
||||
xlim(popt[1]-2*FWHM, popt[1]+2*FWHM)
|
||||
|
||||
show()
|
||||
self.assertEqual(FWHM, 0.12975107355013618)
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
|
||||
@@ -1,201 +1,209 @@
|
||||
#!/usr/bin/env python
|
||||
# -*- coding: utf-8 -*-
|
||||
"""Numerical solver.
|
||||
|
||||
:Date: 2019-05-25
|
||||
|
||||
.. module:: solver
|
||||
:platform: *nix, Windows
|
||||
:synopsis: Numerical solver.
|
||||
|
||||
.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
|
||||
"""
|
||||
from __future__ import division, print_function
|
||||
import unittest
|
||||
import math
|
||||
from numpy import linspace, allclose
|
||||
from numpy.linalg import inv
|
||||
|
||||
import os
|
||||
import sys
|
||||
sys.path.insert(0, os.path.abspath('../src'))
|
||||
import solver
|
||||
import solver_model
|
||||
|
||||
#### Post processing
|
||||
from matplotlib.pyplot import figure, subplots, plot, show
|
||||
from numpy import sqrt, square, concatenate
|
||||
|
||||
def plotx1rphi(x, t, title):
|
||||
"""Plot plane rotation (x, y, phi)
|
||||
"""
|
||||
x1 = x[:, 0]
|
||||
x2 = x[:, 1]
|
||||
x3 = x[:, 2]
|
||||
xp1 = x[:, 3]
|
||||
xp2 = x[:, 4]
|
||||
xp3 = x[:, 5]
|
||||
fig, ax1 = subplots()
|
||||
ax1.set_title(title)
|
||||
ax1.plot(t, x1, 'b-')
|
||||
#ax1.plot(t, x3, 'b-')
|
||||
ax1.set_xlabel('time t in s')
|
||||
# Make the y-axis label and tick labels match the line color.
|
||||
ax1.set_ylabel('x_1', color='b')
|
||||
for tl in ax1.get_yticklabels():
|
||||
tl.set_color('b')
|
||||
envelope = sqrt(square(x1)+square(x2))
|
||||
ax1.plot(t, envelope, 'b--', t, -envelope, 'b--')
|
||||
ax2 = ax1.twinx()
|
||||
ax2.plot(t, xp3, 'r')
|
||||
ax2.set_ylabel('phi', color='r')
|
||||
for tl in ax2.get_yticklabels():
|
||||
tl.set_color('r')
|
||||
if max(xp3) < 2:
|
||||
ax2.set_ylim([0,2])
|
||||
|
||||
f = solver_model.disk
|
||||
fnm = solver_model.disk_nm
|
||||
fnmmdk = solver_model.disk_nmmdk
|
||||
|
||||
|
||||
class TestSolver(unittest.TestCase):
|
||||
|
||||
Dt = .01 # Delta t
|
||||
tend = .1 # t_end
|
||||
#tend = 300
|
||||
nt = math.ceil(tend/Dt) # number of timesteps
|
||||
t = linspace(0., tend, nt+1) # time vector
|
||||
x0 = 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 # initial conditions
|
||||
p = .01, .002, .015 # parameter of the model
|
||||
|
||||
def test_e1(self):
|
||||
xe1 = solver.e1(f, self.x0, self.t, *self.p)
|
||||
#plotx1rphi(xe1, self.t, ("Euler (Runge-Kutta 1th order), increments: %d" % self.nt))
|
||||
#show()
|
||||
self.assertTrue(allclose(xe1[:, 5],
|
||||
[0., 0.00015, 0.0003, 0.00045, 0.0006, 0.00075, 0.0009, 0.00105, 0.0012,
|
||||
0.00135, 0.0015 ]))
|
||||
|
||||
Dt_2 = Dt/5.
|
||||
nt_2 = math.ceil(tend/Dt_2)
|
||||
t_2 = linspace(0., tend, nt_2+1)
|
||||
|
||||
def test_e1_2(self):
|
||||
xe1_2 = solver.e1(f, self.x0, self.t_2, *self.p)
|
||||
#plotx1rphi(xe1_2, self.t_2, ("Euler (Runge-Kutta 1th order), increments: %d" % self.nt_2))
|
||||
#show()
|
||||
self.assertTrue(allclose(xe1_2[:, 5],
|
||||
[0.00000000e+00, 3.00000000e-05, 6.00000000e-05, 8.99999998e-05,
|
||||
1.19999999e-04, 1.49999998e-04, 1.79999995e-04, 2.09999992e-04,
|
||||
2.39999987e-04, 2.69999980e-04, 2.99999971e-04, 3.29999960e-04,
|
||||
3.59999947e-04, 3.89999931e-04, 4.19999913e-04, 4.49999891e-04,
|
||||
4.79999866e-04, 5.09999837e-04, 5.39999804e-04, 5.69999767e-04,
|
||||
5.99999726e-04, 6.29999681e-04, 6.59999630e-04, 6.89999575e-04,
|
||||
7.19999514e-04, 7.49999448e-04, 7.79999376e-04, 8.09999298e-04,
|
||||
8.39999214e-04, 8.69999123e-04, 8.99999026e-04, 9.29998921e-04,
|
||||
9.59998810e-04, 9.89998691e-04, 1.01999856e-03, 1.04999843e-03,
|
||||
1.07999829e-03, 1.10999814e-03, 1.13999798e-03, 1.16999781e-03,
|
||||
1.19999763e-03, 1.22999744e-03, 1.25999725e-03, 1.28999704e-03,
|
||||
1.31999682e-03, 1.34999660e-03, 1.37999636e-03, 1.40999611e-03,
|
||||
1.43999585e-03, 1.46999558e-03, 1.49999530e-03]
|
||||
))
|
||||
|
||||
maxIterations = 1000
|
||||
tol = 1e-9 # against residuum
|
||||
|
||||
def test_e2(self):
|
||||
xe2 = solver.e2(f, self.x0, self.t, *self.p)
|
||||
#plotx1rphi(xe2, self.t, ("Runge-Kutta 2th order, increments: %d" % self.nt))
|
||||
#show()
|
||||
self.assertTrue(allclose(xe2[:, 5],
|
||||
[0., 0.00015, 0.0003, 0.00045, 0.0006, 0.00075, 0.0009, 0.00105, 0.0012,
|
||||
0.00135, 0.0015 ]))
|
||||
|
||||
def test_e4(self):
|
||||
xe4 = solver.e4(f, self.x0, self.t, *self.p)
|
||||
#plotx1rphi(xe4, self.t, ("Runge-Kutta 4th order, increments: %d" % self.nt))
|
||||
#show()
|
||||
self.assertTrue(allclose(xe4[:, 5],
|
||||
[0., 0.00015, 0.0003, 0.00045, 0.0006, 0.00075, 0.0009, 0.00105, 0.0012,
|
||||
0.00135, 0.0015 ]))
|
||||
|
||||
def test_i1(self):
|
||||
xi1, iterations = solver.i1(f, self.x0, self.t, *self.p, self.maxIterations, self.tol)
|
||||
#plotx1rphi(xi1, self.t, ("Backward Euler, increments: %d, iterations: %d" % (self.nt, sum(iterations))))
|
||||
#show()
|
||||
self.assertTrue(allclose(xi1[:, 5],
|
||||
[0., 0.00015, 0.0003, 0.00045, 0.0006, 0.00075, 0.0009, 0.00105, 0.0012,
|
||||
0.00135, 0.0015 ]))
|
||||
|
||||
gamma = .5 # newmark parameter for velocity
|
||||
beta = .25 # newmark parameter for displacement
|
||||
|
||||
def test_nm(self):
|
||||
xnm, xpnm, xppnm, iterations = solver.newmark_newtonraphson(fnm,
|
||||
(0,0,0), (0,0,0), (0,0,0), self.t, *self.p,
|
||||
self.gamma, self.beta, 1, self.tol)
|
||||
#print(xnm)
|
||||
#print(xpnm)
|
||||
x = concatenate((xnm, xpnm), axis=1)
|
||||
#plotx1rphi(x, self.t, ("Newmark, increments: %d, iterations: %d" % (self.nt, sum(iterations))))
|
||||
#show()
|
||||
self.assertTrue(allclose(x[:, 5],
|
||||
[0.00000000e+00, 7.49999925e-05, 2.24999951e-04, 3.74999839e-04,
|
||||
5.24999621e-04, 6.74999269e-04, 8.24998752e-04, 9.74998039e-04,
|
||||
1.12499710e-03, 1.27499591e-03, 1.42499444e-03]
|
||||
))
|
||||
|
||||
def test_nm_2(self):
|
||||
xnm, xpnm, xppnm, iterations = solver.newmark_newtonraphson(fnm,
|
||||
(0,0,0), (0,0,0), (0,0,0), self.t, *self.p,
|
||||
self.gamma, self.beta, self.maxIterations, self.tol)
|
||||
#print(xnm)
|
||||
#print(xpnm)
|
||||
x = concatenate((xnm, xpnm), axis=1)
|
||||
#plotx1rphi(x, self.t, ("Newmark, increments: %d, iterations: %d" % (self.nt, sum(iterations))))
|
||||
#show()
|
||||
self.assertTrue(allclose(x[:, 5],
|
||||
[0.00000000e+00, 7.49999925e-05, 2.24999951e-04, 3.74999839e-04,
|
||||
5.24999621e-04, 6.74999269e-04, 8.24998752e-04, 9.74998039e-04,
|
||||
1.12499710e-03, 1.27499591e-03, 1.42499444e-03]
|
||||
))
|
||||
|
||||
def test_nmmdk(self):
|
||||
xnm, xpnm, xppnm, iterations = solver.newmark_newtonraphson_rdk(fnmmdk,
|
||||
(0,0,0), (0,0,0), (0,0,0), self.t, *self.p,
|
||||
self.gamma, self.beta, 1, self.tol)
|
||||
#print(xnm)
|
||||
#print(xpnm)
|
||||
x = concatenate((xnm, xpnm), axis=1)
|
||||
#plotx1rphi(x, self.t, ("Newmark, increments: %d, iterations: %d" % (self.nt, sum(iterations))))
|
||||
#show()
|
||||
self.assertTrue(allclose(x[:, 5],
|
||||
[0.00000000e+00, 7.49999963e-05, 2.24999974e-04, 3.74999906e-04,
|
||||
5.24999764e-04, 6.74999516e-04, 8.24999134e-04, 9.74998586e-04,
|
||||
1.12499784e-03, 1.27499688e-03, 1.42499566e-03]
|
||||
))
|
||||
|
||||
def test_nmmdk_2(self):
|
||||
xnm, xpnm, xppnm, iterations = solver.newmark_newtonraphson_rdk(fnmmdk,
|
||||
(0,0,0), (0,0,0), (0,0,0), self.t, *self.p,
|
||||
self.gamma, self.beta, 100, self.tol)
|
||||
#print(xnm)
|
||||
#print(xpnm)
|
||||
x = concatenate((xnm, xpnm), axis=1)
|
||||
#plotx1rphi(x, self.t, ("Newmark, increments: %d, iterations: %d" % (self.nt, sum(iterations))))
|
||||
#show()
|
||||
self.assertTrue(allclose(x[:, 5],
|
||||
[0.00000000e+00, 7.49999963e-05, 2.24999974e-04, 3.74999906e-04,
|
||||
5.24999764e-04, 6.74999516e-04, 8.24999134e-04, 9.74998586e-04,
|
||||
1.12499784e-03, 1.27499688e-03, 1.42499566e-03]
|
||||
))
|
||||
|
||||
#from scipy.integrate import odeint
|
||||
#xode = odeint(solver_model.disk, x0, t, p, printmessg=True)
|
||||
#plotx1rphi(xode, t, ("ODE, Inkremente: %d" % nt))
|
||||
#show()
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
unittest.main()
|
||||
#!/usr/bin/env python
|
||||
# -*- coding: utf-8 -*-
|
||||
"""Test of numerical solver.
|
||||
|
||||
:Date: 2019-05-25
|
||||
|
||||
.. module:: test_ode
|
||||
:platform: *nix, Windows
|
||||
:synopsis: Test of numerical solver.
|
||||
|
||||
.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
|
||||
"""
|
||||
from __future__ import division, print_function
|
||||
import unittest
|
||||
import math
|
||||
from numpy import linspace, allclose, sqrt, square, concatenate
|
||||
from numpy.linalg import inv
|
||||
from matplotlib.pyplot import figure, subplots, plot, show
|
||||
|
||||
import os
|
||||
import sys
|
||||
sys.path.insert(0, os.path.abspath('../src'))
|
||||
from numerical.ode import (e1, e2, e4, i1, newmark_newtonraphson,
|
||||
newmark_newtonraphson_rdk)
|
||||
from numerical.ode_model import disk, disk_nm, disk_nmmdk
|
||||
|
||||
def plotx1rphi(x, t, title):
|
||||
"""Plot plane rotation (x, y, phi)
|
||||
"""
|
||||
x1 = x[:, 0]
|
||||
x2 = x[:, 1]
|
||||
x3 = x[:, 2]
|
||||
xp1 = x[:, 3]
|
||||
xp2 = x[:, 4]
|
||||
xp3 = x[:, 5]
|
||||
fig, ax1 = subplots()
|
||||
ax1.set_title(title)
|
||||
ax1.plot(t, x1, 'b-')
|
||||
#ax1.plot(t, x3, 'b-')
|
||||
ax1.set_xlabel('time t in s')
|
||||
# Make the y-axis label and tick labels match the line color.
|
||||
ax1.set_ylabel('x_1', color='b')
|
||||
for tl in ax1.get_yticklabels():
|
||||
tl.set_color('b')
|
||||
envelope = sqrt(square(x1)+square(x2))
|
||||
ax1.plot(t, envelope, 'b--', t, -envelope, 'b--')
|
||||
ax2 = ax1.twinx()
|
||||
ax2.plot(t, xp3, 'r')
|
||||
ax2.set_ylabel('phi', color='r')
|
||||
for tl in ax2.get_yticklabels():
|
||||
tl.set_color('r')
|
||||
if max(xp3) < 2:
|
||||
ax2.set_ylim([0,2])
|
||||
|
||||
f = disk
|
||||
fnm = disk_nm
|
||||
fnmmdk = disk_nmmdk
|
||||
|
||||
|
||||
class TestODE(unittest.TestCase):
|
||||
|
||||
Dt = .01 # Delta t
|
||||
tend = .1 # t_end
|
||||
#tend = 300
|
||||
nt = math.ceil(tend/Dt) # number of timesteps
|
||||
t = linspace(0., tend, nt+1) # time vector
|
||||
x0 = 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 # initial conditions
|
||||
p = .01, .002, .015 # parameter of the model
|
||||
|
||||
def test_e1(self):
|
||||
xe1 = e1(f, self.x0, self.t, *self.p)
|
||||
#plotx1rphi(xe1, self.t, ("Euler (Runge-Kutta 1th order),
|
||||
# increments: %d" % self.nt))
|
||||
#show()
|
||||
self.assertTrue(allclose(xe1[:, 5],
|
||||
[0., 0.00015, 0.0003, 0.00045, 0.0006, 0.00075, 0.0009,
|
||||
0.00105, 0.0012, 0.00135, 0.0015 ]))
|
||||
|
||||
Dt_2 = Dt/5.
|
||||
nt_2 = math.ceil(tend/Dt_2)
|
||||
t_2 = linspace(0., tend, nt_2+1)
|
||||
|
||||
def test_e1_2(self):
|
||||
xe1_2 = e1(f, self.x0, self.t_2, *self.p)
|
||||
#plotx1rphi(xe1_2, self.t_2, ("Euler (Runge-Kutta 1th order),
|
||||
# increments: %d" % self.nt_2))
|
||||
#show()
|
||||
self.assertTrue(allclose(xe1_2[:, 5],
|
||||
[0.00000000e+00, 3.00000000e-05, 6.00000000e-05, 8.99999998e-05,
|
||||
1.19999999e-04, 1.49999998e-04, 1.79999995e-04, 2.09999992e-04,
|
||||
2.39999987e-04, 2.69999980e-04, 2.99999971e-04, 3.29999960e-04,
|
||||
3.59999947e-04, 3.89999931e-04, 4.19999913e-04, 4.49999891e-04,
|
||||
4.79999866e-04, 5.09999837e-04, 5.39999804e-04, 5.69999767e-04,
|
||||
5.99999726e-04, 6.29999681e-04, 6.59999630e-04, 6.89999575e-04,
|
||||
7.19999514e-04, 7.49999448e-04, 7.79999376e-04, 8.09999298e-04,
|
||||
8.39999214e-04, 8.69999123e-04, 8.99999026e-04, 9.29998921e-04,
|
||||
9.59998810e-04, 9.89998691e-04, 1.01999856e-03, 1.04999843e-03,
|
||||
1.07999829e-03, 1.10999814e-03, 1.13999798e-03, 1.16999781e-03,
|
||||
1.19999763e-03, 1.22999744e-03, 1.25999725e-03, 1.28999704e-03,
|
||||
1.31999682e-03, 1.34999660e-03, 1.37999636e-03, 1.40999611e-03,
|
||||
1.43999585e-03, 1.46999558e-03, 1.49999530e-03]
|
||||
))
|
||||
|
||||
maxIterations = 1000
|
||||
tol = 1e-9 # against residuum
|
||||
|
||||
def test_e2(self):
|
||||
xe2 = e2(f, self.x0, self.t, *self.p)
|
||||
#plotx1rphi(xe2, self.t, ("Runge-Kutta 2th order,
|
||||
# increments: %d" % self.nt))
|
||||
#show()
|
||||
self.assertTrue(allclose(xe2[:, 5],
|
||||
[0., 0.00015, 0.0003, 0.00045, 0.0006, 0.00075, 0.0009,
|
||||
0.00105, 0.0012, 0.00135, 0.0015 ]))
|
||||
|
||||
def test_e4(self):
|
||||
xe4 = e4(f, self.x0, self.t, *self.p)
|
||||
#plotx1rphi(xe4, self.t, ("Runge-Kutta 4th order,
|
||||
# increments: %d" % self.nt))
|
||||
#show()
|
||||
self.assertTrue(allclose(xe4[:, 5],
|
||||
[0., 0.00015, 0.0003, 0.00045, 0.0006, 0.00075, 0.0009,
|
||||
0.00105, 0.0012, 0.00135, 0.0015 ]))
|
||||
|
||||
def test_i1(self):
|
||||
xi1, iterations = i1(f, self.x0, self.t, *self.p,
|
||||
self.maxIterations, self.tol)
|
||||
#plotx1rphi(xi1, self.t, ("Backward Euler, increments: %d,
|
||||
# iterations: %d" % (self.nt, sum(iterations))))
|
||||
#show()
|
||||
self.assertTrue(allclose(xi1[:, 5],
|
||||
[0., 0.00015, 0.0003, 0.00045, 0.0006, 0.00075, 0.0009,
|
||||
0.00105, 0.0012, 0.00135, 0.0015 ]))
|
||||
|
||||
gamma = .5 # newmark parameter for velocity
|
||||
beta = .25 # newmark parameter for displacement
|
||||
|
||||
def test_nm(self):
|
||||
xnm, xpnm, xppnm, iterations = newmark_newtonraphson(fnm,
|
||||
(0,0,0), (0,0,0), (0,0,0), self.t, *self.p,
|
||||
self.gamma, self.beta, 1, self.tol)
|
||||
#print(xnm)
|
||||
#print(xpnm)
|
||||
x = concatenate((xnm, xpnm), axis=1)
|
||||
#plotx1rphi(x, self.t, ("Newmark, increments: %d,
|
||||
# iterations: %d" % (self.nt, sum(iterations))))
|
||||
#show()
|
||||
self.assertTrue(allclose(x[:, 5],
|
||||
[0.00000000e+00, 7.49999925e-05, 2.24999951e-04, 3.74999839e-04,
|
||||
5.24999621e-04, 6.74999269e-04, 8.24998752e-04, 9.74998039e-04,
|
||||
1.12499710e-03, 1.27499591e-03, 1.42499444e-03]
|
||||
))
|
||||
|
||||
def test_nm_2(self):
|
||||
xnm, xpnm, xppnm, iterations = newmark_newtonraphson(fnm,
|
||||
(0,0,0), (0,0,0), (0,0,0), self.t, *self.p,
|
||||
self.gamma, self.beta, self.maxIterations, self.tol)
|
||||
#print(xnm)
|
||||
#print(xpnm)
|
||||
x = concatenate((xnm, xpnm), axis=1)
|
||||
#plotx1rphi(x, self.t, ("Newmark, increments: %d,
|
||||
# iterations: %d" % (self.nt, sum(iterations))))
|
||||
#show()
|
||||
self.assertTrue(allclose(x[:, 5],
|
||||
[0.00000000e+00, 7.49999925e-05, 2.24999951e-04, 3.74999839e-04,
|
||||
5.24999621e-04, 6.74999269e-04, 8.24998752e-04, 9.74998039e-04,
|
||||
1.12499710e-03, 1.27499591e-03, 1.42499444e-03]
|
||||
))
|
||||
|
||||
def test_nmmdk(self):
|
||||
xnm, xpnm, xppnm, iterations = newmark_newtonraphson_rdk(
|
||||
fnmmdk, (0,0,0), (0,0,0), (0,0,0), self.t, *self.p,
|
||||
self.gamma, self.beta, 1, self.tol)
|
||||
#print(xnm)
|
||||
#print(xpnm)
|
||||
x = concatenate((xnm, xpnm), axis=1)
|
||||
#plotx1rphi(x, self.t, ("Newmark, increments: %d,
|
||||
# iterations: %d" % (self.nt, sum(iterations))))
|
||||
#show()
|
||||
self.assertTrue(allclose(x[:, 5],
|
||||
[0.00000000e+00, 7.49999963e-05, 2.24999974e-04, 3.74999906e-04,
|
||||
5.24999764e-04, 6.74999516e-04, 8.24999134e-04, 9.74998586e-04,
|
||||
1.12499784e-03, 1.27499688e-03, 1.42499566e-03]
|
||||
))
|
||||
|
||||
def test_nmmdk_2(self):
|
||||
xnm, xpnm, xppnm, iterations = newmark_newtonraphson_rdk(
|
||||
fnmmdk, (0,0,0), (0,0,0), (0,0,0), self.t, *self.p,
|
||||
self.gamma, self.beta, 100, self.tol)
|
||||
#print(xnm)
|
||||
#print(xpnm)
|
||||
x = concatenate((xnm, xpnm), axis=1)
|
||||
#plotx1rphi(x, self.t, ("Newmark, increments: %d,
|
||||
# iterations: %d" % (self.nt, sum(iterations))))
|
||||
#show()
|
||||
self.assertTrue(allclose(x[:, 5],
|
||||
[0.00000000e+00, 7.49999963e-05, 2.24999974e-04, 3.74999906e-04,
|
||||
5.24999764e-04, 6.74999516e-04, 8.24999134e-04, 9.74998586e-04,
|
||||
1.12499784e-03, 1.27499688e-03, 1.42499566e-03]
|
||||
))
|
||||
|
||||
#from scipy.integrate import odeint
|
||||
#xode = odeint(disk, x0, t, p, printmessg=True)
|
||||
#plotx1rphi(xode, t, ("ODE, Inkremente: %d" % nt))
|
||||
#show()
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
unittest.main()
|
||||
Reference in New Issue
Block a user