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00 Basic introduction to Python for scientific calculations.html
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00 Basic introduction to Python for scientific calculations.html
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<!DOCTYPE html>
<html>
<head><meta charset="utf-8" />
<title>00 Basic introduction to Python for scientific calculations</title>
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<h1 id="Basic-introduction-to-Python-for-scientific-calculations">Basic introduction to Python for scientific calculations<a class="anchor-link" href="#Basic-introduction-to-Python-for-scientific-calculations">¶</a></h1>
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<p>Python is organised in packages, modules and functions in that hierarchical order.</p>
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<p>The most important Python packages for scientific calculations are <strong>numpy</strong> and <strong>scipy</strong>. The basic package for generating scientific plots is <strong>matplotlib</strong>.</p>
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<p>Python packages typically contain several modules, where each module is a separate file with file ending ".py". For instance, <strong>scipy</strong> contains the module <strong>stats</strong> and <strong>matplotlib</strong> the module <strong>pyplot</strong>.</p>
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<p>Each module can contain several functions.</p>
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<p>A Python program starts with an import of the required modules such as</p>
<div class="highlight"><pre><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="kn">import</span> <span class="n">normal</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
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<ul>
<li>When a function is imported (such as <em>normal</em> above) it can be used by its name.</li>
<li>When a module is imported, then its functions are used by appending a "." and the function's name to the module name<div class="highlight"><pre><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span>
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<li>When the sought function is located in a sub-module, the "." notation can be used again<div class="highlight"><pre><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">lstsq</span><span class="p">()</span>
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<p>Functions in Python can have necessary and optional arguments. Optional arguments are defined by keyword, that is their name in the function call.</p>
<div class="highlight"><pre><span class="n">normal</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">loc</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">scale</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span>
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<p>Keyword arguments are used to define default values for arguments that can be changed by the user.</p>
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<h3 id="Vectors-and-matrices">Vectors and matrices<a class="anchor-link" href="#Vectors-and-matrices">¶</a></h3><p>The Python package <strong>numpy</strong> contains several powerful tools for efficient matrix-vector algebra.</p>
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<p>A vector in Python is a numpy array:</p>
<div class="highlight"><pre><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="n">vec</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</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="p">)</span>
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<p>Similarly, a matrix is a two-dimensional numpy array:</p>
<div class="highlight"><pre><span class="n">mat</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[</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="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="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">27</span><span class="p">]</span> <span class="p">]</span> <span class="p">)</span>
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<p>Calculations with matrices and vectors are implemented easily:</p>
<ul>
<li><p>addition and subtraction</p>
<div class="highlight"><pre><span class="n">vec</span> <span class="o">+</span> <span class="n">vec</span>
<span class="n">vec</span> <span class="o">-</span> <span class="n">vec</span>
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<li><p>element-wise multiplication and division</p>
<div class="highlight"><pre><span class="n">vec</span> <span class="o">*</span> <span class="n">vec</span>
<span class="n">vec</span> <span class="o">/</span> <span class="n">vec</span>
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<li><p>matrix transpose</p>
<div class="highlight"><pre><span class="n">transposed</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">T</span>
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<p>Special vector and matrix algebra are built-in numpy functions</p>
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<li><p>inner product</p>
<div class="highlight"><pre><span class="n">vec1</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">vec</span> <span class="p">)</span>
<span class="n">mat</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span> <span class="n">vec</span> <span class="p">)</span>
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<li><p>matrix inverse</p>
<div class="highlight"><pre><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inverse</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
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<li><p>product of matrix inverse and another matrix</p>
<div class="highlight"><pre><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">mat1</span><span class="p">,</span> <span class="n">mat2</span><span class="p">)</span>
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<li><p>matrix rank</p>
<div class="highlight"><pre><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_rank</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
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<p>Indexing into a vector</p>
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<li>indexing in Python starts at $0$</li>
<li>ranges can be obtained using $:$</li>
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<div class="highlight"><pre><span class="n">v</span> <span class="o">=</span> <span class="n">np</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">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">])</span>
<span class="n">v</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="mi">4</span>
<span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="mi">5</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</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="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
<span class="n">v</span><span class="p">[:</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</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">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
<span class="n">v</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">6</span>
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<li><p>matrices are indexed as</p>
<div class="highlight"><pre><span class="n">mat</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">mat</span><span class="p">[:</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">:</span><span class="mi">4</span><span class="p">]</span>
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<li><p>fancy indexing also works</p>
<div class="highlight"><pre><span class="n">v</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</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="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
<span class="n">w</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">11</span><span class="p">,</span><span class="mi">12</span><span class="p">,</span><span class="mi">13</span><span class="p">,</span><span class="mi">14</span><span class="p">])</span>
<span class="n">w</span><span class="p">[</span><span class="n">v</span><span class="o">></span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">13</span><span class="p">,</span><span class="mi">14</span><span class="p">])</span>
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<p>Some special arrays can be generated using built-in numpy functions</p>
<ul>
<li>array containing all zeros or ones<div class="highlight"><pre><span class="n">mat</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span> <span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> <span class="p">)</span>
<span class="n">vec</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
<span class="n">complex_mat</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span> <span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">20</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">complex</span> <span class="p">)</span>
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<h3 id="Plotting">Plotting<a class="anchor-link" href="#Plotting">¶</a></h3><p>The <strong>matplotlib.pyplot</strong> package contains several functions for basic plotting tasks. Its syntax is very similar to that of Matlab.</p>
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<div class="highlight"><pre><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">time</span><span class="p">,</span> <span class="n">signal</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"my signal"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"upper left"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"x axis"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"y axis"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">signal</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"histogram of values in signal"</span><span class="p">)</span>
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