【编程】学习Python数学库Tentative NumPy Tutorial - 科技一览 - 同义词网 - Powered by Discuz! Archiver

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jason246 发表于 2023-6-4 08:36:06

【编程】学习Python数学库Tentative NumPy Tutorial

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<div class="table-of-contents" style="padding:1em; margin:1em; max-width:100%;"><p class="table-of-contents-heading">Contents</p><ol><li>
<a href="#Prerequisites">Prerequisites</a></li><li>
<a href="#The_Basics">The Basics</a><ol><li>
<a href="#An_example">An example</a></li><li>
<a href="#Array_Creation">Array Creation</a></li><li>
<a href="#Printing_Arrays">Printing Arrays</a></li><li>
<a href="#Basic_Operations">Basic Operations</a></li><li>
<a href="#Universal_Functions">Universal Functions</a></li><li>
<a href="#Indexing.2C_Slicing_and_Iterating">Indexing, Slicing and Iterating</a></li></ol></li><li>
<a href="#Shape_Manipulation">Shape Manipulation</a><ol><li>
<a href="#Changing_the_shape_of_an_array">Changing the shape of an array</a></li><li>
<a href="#Stacking_together_different_arrays">Stacking together different arrays</a></li><li>
<a href="#Splitting_one_array_into_several_smaller_ones">Splitting one array into several smaller ones</a></li></ol></li><li>
<a href="#Copies_and_Views">Copies and Views</a><ol><li>
<a href="#No_Copy_at_All">No Copy at All</a></li><li>
<a href="#View_or_Shallow_Copy">View or Shallow Copy</a></li><li>
<a href="#Deep_Copy">Deep Copy</a></li><li>
<a href="#Functions_and_Methods_Overview">Functions and Methods Overview</a></li></ol></li><li>
<a href="#Less_Basic">Less Basic</a><ol><li>
<a href="#Broadcasting_rules">Broadcasting rules</a></li></ol></li><li>
<a href="#Fancy_indexing_and_index_tricks">Fancy indexing and index tricks</a><ol><li>
<a href="#Indexing_with_Arrays_of_Indices">Indexing with Arrays of Indices</a></li><li>
<a href="#Indexing_with_Boolean_Arrays">Indexing with Boolean Arrays</a></li><li>
<a href="#The_ix_.28.29_function">The ix_() function</a></li><li>
<a href="#Indexing_with_strings">Indexing with strings</a></li></ol></li><li>
<a href="#Linear_Algebra">Linear Algebra</a><ol><li>
<a href="#Simple_Array_Operations">Simple Array Operations</a></li><li>
<a href="#The_Matrix_Class">The Matrix Class</a></li><li>
<a href="#Indexing:_Comparing_Matrices_and_2D_Arrays">Indexing: Comparing Matrices and 2D Arrays</a></li></ol></li><li>
<a href="#Tricks_and_Tips">Tricks and Tips</a><ol><li>
<a href="#A.22Automatic.22_Reshaping">"Automatic" Reshaping</a></li><li>
<a href="#Vector_Stacking">Vector Stacking</a></li><li>
<a href="#Histograms">Histograms</a></li></ol></li><li>
<a href="#References">References</a></li></ol></div> <span class="anchor" id="line-4"></span><span class="anchor" id="line-5"></span><p class="line867">
</p><h2 id="Prerequisites">Prerequisites</h2>
<span class="anchor" id="line-6"></span><p class="line862">Before reading this tutorial you should know a bit of Python. If you would like to refresh your memory, take a look at the <ahref="http://docs.python.org/tut/">Python tutorial</a>. <span class="anchor" id="line-7"></span><span class="anchor" id="line-8"></span></p><p class="line874">If you wish to work the examples in this tutorial, you must also have some software installed on your computer. Minimally: <span class="anchor" id="line-9"></span></p><ul><li><p class="line891"><ahref="http://www.python.org/">Python</a> <span class="anchor" id="line-10"></span></p></li><li><p class="line891"><ahref="http://numpy.scipy.org/">NumPy</a> <span class="anchor" id="line-11"></span><span class="anchor" id="line-12"></span></p></li></ul><p class="line874">These you may find useful: <span class="anchor" id="line-13"></span></p><ul><li><p class="line891"><ahref="http://ipython.scipy.org/">ipython</a> is an enhanced interactive Python shell which is very convenient for exploring NumPy's features <span class="anchor" id="line-14"></span></p></li><li><p class="line891"><ahref="http://matplotlib.sourceforge.net/">matplotlib</a> will enable you to plot graphics <span class="anchor" id="line-15"></span></p></li><li><p class="line891"><ahref="http://scipy.org/">SciPy</a> provides a lot of scientific routines that work on top of NumPy <span class="anchor" id="line-16"></span><span class="anchor" id="line-17"></span></p></li></ul><p class="line867">
</p><h2 id="The_Basics">The Basics</h2>
<span class="anchor" id="line-18"></span><p class="line867">NumPy's main object is the homogeneous multidimensional array. It is a table of elements (usually numbers), all of the same type, indexed by a tuple of positive integers. In Numpy dimensions are called <em>axes</em>.The number of axes is <em>rank</em>. <span class="anchor" id="line-19"></span><span class="anchor" id="line-20"></span></p><p class="line862">For example, the coordinates of a point in 3D space <tt class="backtick"></tt> is an array of rank 1, because it has one axis. That axis has a length of 3. <span class="anchor" id="line-21"></span>In example pictured below, the array has rank 2 (it is 2-dimensional). The first dimension (axis) has a length of 2, the second dimension has a length of 3. <span class="anchor" id="line-22"></span><span class="anchor" id="line-23"></span></p><p class="line867"><span class="anchor" id="line-24"></span><span class="anchor" id="line-25"></span><span class="anchor" id="line-26"></span></p><pre><span class="anchor" id="line-1"></span>[[ 1., 0., 0.],
<span class="anchor" id="line-2"></span> [ 0., 1., 2.]]</pre><span class="anchor" id="line-27"></span><span class="anchor" id="line-28"></span><span class="anchor" id="line-29"></span><p class="line862">Numpy's array class is called <tt class="backtick">ndarray</tt>. It is also known by the alias <tt class="backtick">array</tt>. Note that <tt class="backtick">numpy.array</tt> is not the same as the Standard Python Library class <tt class="backtick">array.array</tt>, which only handles one-dimensional arrays and offers less functionality. The more important attributes of an <tt class="backtick">ndarray</tt> object are: <span class="anchor" id="line-30"></span><span class="anchor" id="line-31"></span></p><dl><dt>ndarray.ndim</dt><dd><p class="line862">the number of axes (dimensions) of the array. In the Python world, the number of dimensions is referred to as <em>rank</em>. <span class="anchor" id="line-32"></span></p></dd><dt>ndarray.shape</dt><dd><p class="line862">the dimensions of the array. This is a tuple of integers indicating the size of the array in each dimension. For a matrix with <em>n</em> rows and <em>m</em> columns, <tt class="backtick">shape</tt> will be <tt class="backtick">(n,m)</tt>. The length of the <tt class="backtick">shape</tt> tuple is therefore the rank, or number of dimensions, <tt class="backtick">ndim</tt>. <span class="anchor" id="line-33"></span></p></dd><dt>ndarray.size</dt><dd><p class="line862">the total number of elements of the array. This is equal to the product of the elements of <tt class="backtick">shape</tt>. <span class="anchor" id="line-34"></span></p></dd><dt>ndarray.dtype</dt><dd><p class="line862">an object describing the type of the elements in the array. One can create or specify dtype's using standard Python types. Additionally NumPy provides types of its own.numpy.int32, numpy.int16, and numpy.float64 are some examples. <span class="anchor" id="line-35"></span></p></dd><dt>ndarray.itemsize</dt><dd><p class="line862">the size in bytes of each element of the array. For example, an array of elements of type <tt class="backtick">float64</tt> has <tt class="backtick">itemsize</tt> 8 (=64/8), while one of type <tt class="backtick">complex32</tt> has <tt class="backtick">itemsize</tt> 4 (=32/8). It is equivalent to <tt class="backtick">ndarray.dtype.itemsize</tt>. <span class="anchor" id="line-36"></span></p></dd><dt>ndarray.data</dt><dd>the buffer containing the actual elements of the array. Normally, we won't need to use this attribute because we will access the elements in an array using indexing facilities. <span class="anchor" id="line-37"></span></dd></dl><p class="line867">
</p><h4 id="An_example">An example</h4>
<span class="anchor" id="line-38"></span><span class="anchor" id="line-39"></span><p class="line867"><span class="anchor" id="line-40"></span><span class="anchor" id="line-41"></span><span class="anchor" id="line-42"></span><span class="anchor" id="line-43"></span><span class="anchor" id="line-44"></span><span class="anchor" id="line-45"></span><span class="anchor" id="line-46"></span><span class="anchor" id="line-47"></span><span class="anchor" id="line-48"></span><span class="anchor" id="line-49"></span><span class="anchor" id="line-50"></span><span class="anchor" id="line-51"></span><span class="anchor" id="line-52"></span><span class="anchor" id="line-53"></span><span class="anchor" id="line-54"></span><span class="anchor" id="line-55"></span><span class="anchor" id="line-56"></span><span class="anchor" id="line-57"></span><span class="anchor" id="line-58"></span><span class="anchor" id="line-59"></span><span class="anchor" id="line-60"></span><span class="anchor" id="line-61"></span><span class="anchor" id="line-62"></span><span class="anchor" id="line-63"></span><span class="anchor" id="line-64"></span><span class="anchor" id="line-1-1"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-dd4a04a85691b2fb2863ea200ba146ce9f15b4df" lang="en"><span class="line"><span class="anchor" id="line-1-2"></span>>>> <span class="ResWord">from</span> <span class="ID">numpy</span><span class="ResWord">import</span> *</span>
<span class="line"><span class="anchor" id="line-2-1"></span>>>> <span class="ID">a</span> = <span class="ID">arange</span>(<span class="Number">15</span>).<span class="ID">reshape</span>(<span class="Number">3</span>, <span class="Number">5</span>)</span>
<span class="line"><span class="anchor" id="line-3-1"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-4-1"></span><span class="ID">array</span>([[ <span class="Number">0</span>,<span class="Number">1</span>,<span class="Number">2</span>,<span class="Number">3</span>,<span class="Number">4</span>],</span>
<span class="line"><span class="anchor" id="line-5-1"></span>    [ <span class="Number">5</span>,<span class="Number">6</span>,<span class="Number">7</span>,<span class="Number">8</span>,<span class="Number">9</span>],</span>
<span class="line"><span class="anchor" id="line-6-1"></span>    [<span class="Number">10</span>, <span class="Number">11</span>, <span class="Number">12</span>, <span class="Number">13</span>, <span class="Number">14</span>]])</span>
<span class="line"><span class="anchor" id="line-7-1"></span>>>> <span class="ID">a</span>.<span class="ID">shape</span></span>
<span class="line"><span class="anchor" id="line-8-1"></span>(<span class="Number">3</span>, <span class="Number">5</span>)</span>
<span class="line"><span class="anchor" id="line-9-1"></span>>>> <span class="ID">a</span>.<span class="ID">ndim</span></span>
<span class="line"><span class="anchor" id="line-10-1"></span><span class="Number">2</span></span>
<span class="line"><span class="anchor" id="line-11-1"></span>>>> <span class="ID">a</span>.<span class="ID">dtype</span>.<span class="ID">name</span></span>
<span class="line"><span class="anchor" id="line-12-1"></span><span class="String">'</span><span class="String">int32</span><span class="String">'</span></span>
<span class="line"><span class="anchor" id="line-13-1"></span>>>> <span class="ID">a</span>.<span class="ID">itemsize</span></span>
<span class="line"><span class="anchor" id="line-14-1"></span><span class="Number">4</span></span>
<span class="line"><span class="anchor" id="line-15-1"></span>>>> <span class="ID">a</span>.<span class="ID">size</span></span>
<span class="line"><span class="anchor" id="line-16-1"></span><span class="Number">15</span></span>
<span class="line"><span class="anchor" id="line-17-1"></span>>>> <span class="ResWord">type</span>(<span class="ID">a</span>)</span>
<span class="line"><span class="anchor" id="line-18-1"></span><span class="ID">numpy</span>.<span class="ID">ndarray</span></span>
<span class="line"><span class="anchor" id="line-19-1"></span>>>> <span class="ID">b</span> = <span class="ID">array</span>([<span class="Number">6</span>, <span class="Number">7</span>, <span class="Number">8</span>])</span>
<span class="line"><span class="anchor" id="line-20-1"></span>>>> <span class="ID">b</span></span>
<span class="line"><span class="anchor" id="line-21-1"></span><span class="ID">array</span>([<span class="Number">6</span>, <span class="Number">7</span>, <span class="Number">8</span>])</span>
<span class="line"><span class="anchor" id="line-22-1"></span>>>> <span class="ResWord">type</span>(<span class="ID">b</span>)</span>
<span class="line"><span class="anchor" id="line-23-1"></span><span class="ID">numpy</span>.<span class="ID">ndarray</span></span>
</pre></div></div><span class="anchor" id="line-65"></span><span class="anchor" id="line-66"></span><span class="anchor" id="line-67"></span><p class="line867">
</p><h4 id="Array_Creation">Array Creation</h4>
<span class="anchor" id="line-68"></span><p class="line874">There are several ways to create arrays. <span class="anchor" id="line-69"></span><span class="anchor" id="line-70"></span></p><p class="line862">For example, you can create an array from a regular Python list or tuple using the <tt class="backtick">array</tt> function. The type of the resulting array is deduced from the type of the elements in the sequences. <span class="anchor" id="line-71"></span><span class="anchor" id="line-72"></span></p><p class="line867"><span class="anchor" id="line-73"></span><span class="anchor" id="line-74"></span><span class="anchor" id="line-75"></span><span class="anchor" id="line-76"></span><span class="anchor" id="line-77"></span><span class="anchor" id="line-78"></span><span class="anchor" id="line-79"></span><span class="anchor" id="line-80"></span><span class="anchor" id="line-81"></span><span class="anchor" id="line-82"></span><span class="anchor" id="line-83"></span><span class="anchor" id="line-1-3"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-8c3dc8b621118a201439f2dcca1770aa1e225f83" lang="en"><span class="line"><span class="anchor" id="line-1-4"></span>>>> <span class="ResWord">from</span> <span class="ID">numpy</span> <span class="ResWord">import</span> *</span>
<span class="line"><span class="anchor" id="line-2-2"></span>>>> <span class="ID">a</span> = <span class="ID">array</span>( [<span class="Number">2</span>,<span class="Number">3</span>,<span class="Number">4</span>] )</span>
<span class="line"><span class="anchor" id="line-3-2"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-4-2"></span><span class="ID">array</span>([<span class="Number">2</span>, <span class="Number">3</span>, <span class="Number">4</span>])</span>
<span class="line"><span class="anchor" id="line-5-2"></span>>>> <span class="ID">a</span>.<span class="ID">dtype</span></span>
<span class="line"><span class="anchor" id="line-6-2"></span><span class="ID">dtype</span>(<span class="String">'</span><span class="String">int32</span><span class="String">'</span>)</span>
<span class="line"><span class="anchor" id="line-7-2"></span>>>> <span class="ID">b</span> = <span class="ID">array</span>([<span class="Number">1.2</span>, <span class="Number">3.5</span>, <span class="Number">5.1</span>])</span>
<span class="line"><span class="anchor" id="line-8-2"></span>>>> <span class="ID">b</span>.<span class="ID">dtype</span></span>
<span class="line"><span class="anchor" id="line-9-2"></span><span class="ID">dtype</span>(<span class="String">'</span><span class="String">float64</span><span class="String">'</span>)</span>
</pre></div></div><span class="anchor" id="line-84"></span><span class="anchor" id="line-85"></span><p class="line862">A frequent error consists in calling <tt class="backtick">array</tt> with multiple numeric arguments, rather than providing a single list of numbers as an argument. <span class="anchor" id="line-86"></span><span class="anchor" id="line-87"></span><span class="anchor" id="line-88"></span><span class="anchor" id="line-89"></span><span class="anchor" id="line-90"></span><span class="anchor" id="line-91"></span><span class="anchor" id="line-1-5"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-79f917c6b2af4e3ee7a4547e576e08ef9e609bc8" lang="en"><span class="line"><span class="anchor" id="line-1-6"></span>>>> <span class="ID">a</span> = <span class="ID">array</span>(<span class="Number">1</span>,<span class="Number">2</span>,<span class="Number">3</span>,<span class="Number">4</span>) <span class="Comment"># WRONG</span></span>
<span class="line"><span class="anchor" id="line-2-3"></span></span>
<span class="line"><span class="anchor" id="line-3-3"></span>>>> <span class="ID">a</span> = <span class="ID">array</span>([<span class="Number">1</span>,<span class="Number">2</span>,<span class="Number">3</span>,<span class="Number">4</span>])<span class="Comment"># RIGHT</span></span>
</pre></div></div><span class="anchor" id="line-92"></span><span class="anchor" id="line-93"></span><p class="line867"><tt class="backtick">array</tt> transforms sequences of sequences into two-dimensional arrays, sequences of sequences of sequences into three-dimensional arrays, and so on. <span class="anchor" id="line-94"></span><span class="anchor" id="line-95"></span></p><p class="line867"><span class="anchor" id="line-96"></span><span class="anchor" id="line-97"></span><span class="anchor" id="line-98"></span><span class="anchor" id="line-99"></span><span class="anchor" id="line-100"></span><span class="anchor" id="line-101"></span><span class="anchor" id="line-1-7"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-3a12d62096acc203ad37f90b8ef77af5abad4dfb" lang="en"><span class="line"><span class="anchor" id="line-1-8"></span>>>> <span class="ID">b</span> = <span class="ID">array</span>( [ (<span class="Number">1.5</span>,<span class="Number">2</span>,<span class="Number">3</span>), (<span class="Number">4</span>,<span class="Number">5</span>,<span class="Number">6</span>) ] )</span>
<span class="line"><span class="anchor" id="line-2-4"></span>>>> <span class="ID">b</span></span>
<span class="line"><span class="anchor" id="line-3-4"></span><span class="ID">array</span>([[ <span class="Number">1.5</span>,<span class="Number">2.</span> ,<span class="Number">3.</span> ],</span>
<span class="line"><span class="anchor" id="line-4-3"></span>    [ <span class="Number">4.</span> ,<span class="Number">5.</span> ,<span class="Number">6.</span> ]])</span>
</pre></div></div><span class="anchor" id="line-102"></span><p class="line874">The type of the array can also be explicitly specified at creation time: <span class="anchor" id="line-103"></span><span class="anchor" id="line-104"></span><span class="anchor" id="line-105"></span><span class="anchor" id="line-106"></span><span class="anchor" id="line-107"></span><span class="anchor" id="line-108"></span><span class="anchor" id="line-109"></span><span class="anchor" id="line-1-9"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-338e7d890db22400eee1a99400d48d71685e0fa4" lang="en"><span class="line"><span class="anchor" id="line-1-10"></span>>>> <span class="ID">c</span> = <span class="ID">array</span>( [ [<span class="Number">1</span>,<span class="Number">2</span>], [<span class="Number">3</span>,<span class="Number">4</span>] ], <span class="ID">dtype</span>=<span class="ResWord">complex</span> )</span>
<span class="line"><span class="anchor" id="line-2-5"></span>>>> <span class="ID">c</span></span>
<span class="line"><span class="anchor" id="line-3-5"></span><span class="ID">array</span>([[ <span class="Number">1.</span>+<span class="Number">0.j</span>,<span class="Number">2.</span>+<span class="Number">0.j</span>],</span>
<span class="line"><span class="anchor" id="line-4-4"></span>    [ <span class="Number">3.</span>+<span class="Number">0.j</span>,<span class="Number">4.</span>+<span class="Number">0.j</span>]])</span>
</pre></div></div><span class="anchor" id="line-110"></span><span class="anchor" id="line-111"></span><p class="line862">Often, the elements of an array are originally unknown, but its size is known. Hence, <a class="nonexistent" href="https://scipy.github.io/old-wiki/pages/NumPy.html">NumPy</a> offers several functions to create arrays with initial placeholder content. These minimize the necessity of growing arrays, an expensive operation. <span class="anchor" id="line-112"></span><span class="anchor" id="line-113"></span></p><p class="line862">The function <tt class="backtick">zeros</tt> creates an array full of zeros, the function <tt class="backtick">ones</tt> creates an array full of ones, and the function <tt class="backtick">empty</tt> creates an array whose initial content is random and depends on the state of the memory. By default, the dtype of the created array is <tt class="backtick">float64</tt>. <span class="anchor" id="line-114"></span><span class="anchor" id="line-115"></span><span class="anchor" id="line-116"></span><span class="anchor" id="line-117"></span><span class="anchor" id="line-118"></span><span class="anchor" id="line-119"></span><span class="anchor" id="line-120"></span><span class="anchor" id="line-121"></span><span class="anchor" id="line-122"></span><span class="anchor" id="line-123"></span><span class="anchor" id="line-124"></span><span class="anchor" id="line-125"></span><span class="anchor" id="line-126"></span><span class="anchor" id="line-127"></span><span class="anchor" id="line-128"></span><span class="anchor" id="line-129"></span><span class="anchor" id="line-130"></span><span class="anchor" id="line-1-11"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-97c9bc2c369a199206490cc6da991428a4880cf0" lang="en"><span class="line"><span class="anchor" id="line-1-12"></span>>>> <span class="ID">zeros</span>( (<span class="Number">3</span>,<span class="Number">4</span>) )</span>
<span class="line"><span class="anchor" id="line-2-6"></span><span class="ID">array</span>([[<span class="Number">0.</span>,<span class="Number">0.</span>,<span class="Number">0.</span>,<span class="Number">0.</span>],</span>
<span class="line"><span class="anchor" id="line-3-6"></span>    [<span class="Number">0.</span>,<span class="Number">0.</span>,<span class="Number">0.</span>,<span class="Number">0.</span>],</span>
<span class="line"><span class="anchor" id="line-4-5"></span>    [<span class="Number">0.</span>,<span class="Number">0.</span>,<span class="Number">0.</span>,<span class="Number">0.</span>]])</span>
<span class="line"><span class="anchor" id="line-5-3"></span>>>> <span class="ID">ones</span>( (<span class="Number">2</span>,<span class="Number">3</span>,<span class="Number">4</span>), <span class="ID">dtype</span>=<span class="ID">int16</span> )             <span class="Comment"># dtype can also be specified</span></span>
<span class="line"><span class="anchor" id="line-6-3"></span><span class="ID">array</span>([[[ <span class="Number">1</span>, <span class="Number">1</span>, <span class="Number">1</span>, <span class="Number">1</span>],</span>
<span class="line"><span class="anchor" id="line-7-3"></span>    [ <span class="Number">1</span>, <span class="Number">1</span>, <span class="Number">1</span>, <span class="Number">1</span>],</span>
<span class="line"><span class="anchor" id="line-8-3"></span>    [ <span class="Number">1</span>, <span class="Number">1</span>, <span class="Number">1</span>, <span class="Number">1</span>]],</span>
<span class="line"><span class="anchor" id="line-9-3"></span>    [[ <span class="Number">1</span>, <span class="Number">1</span>, <span class="Number">1</span>, <span class="Number">1</span>],</span>
<span class="line"><span class="anchor" id="line-10-2"></span>    [ <span class="Number">1</span>, <span class="Number">1</span>, <span class="Number">1</span>, <span class="Number">1</span>],</span>
<span class="line"><span class="anchor" id="line-11-2"></span>    [ <span class="Number">1</span>, <span class="Number">1</span>, <span class="Number">1</span>, <span class="Number">1</span>]]], <span class="ID">dtype</span>=<span class="ID">int16</span>)</span>
<span class="line"><span class="anchor" id="line-12-2"></span>>>> <span class="ID">empty</span>( (<span class="Number">2</span>,<span class="Number">3</span>) )</span>
<span class="line"><span class="anchor" id="line-13-2"></span><span class="ID">array</span>([[<span class="Number">3.73603959e-262</span>, <span class="Number">6.02658058e-154</span>, <span class="Number">6.55490914e-260</span>],</span>
<span class="line"><span class="anchor" id="line-14-2"></span>    [<span class="Number">5.30498948e-313</span>, <span class="Number">3.14673309e-307</span>, <span class="Number">1.00000000e+000</span>]])</span>
</pre></div></div><span class="anchor" id="line-131"></span><span class="anchor" id="line-132"></span><p class="line862">To create sequences of numbers, NumPy provides a function analogous to <tt class="backtick">range</tt> that returns arrays instead of lists <span class="anchor" id="line-133"></span><span class="anchor" id="line-134"></span></p><p class="line867"><span class="anchor" id="line-135"></span><span class="anchor" id="line-136"></span><span class="anchor" id="line-137"></span><span class="anchor" id="line-138"></span><span class="anchor" id="line-139"></span><span class="anchor" id="line-140"></span><span class="anchor" id="line-1-13"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-4689bcd13b952676f290dd7aace2ab084f473ba3" lang="en"><span class="line"><span class="anchor" id="line-1-14"></span>>>> <span class="ID">arange</span>( <span class="Number">10</span>, <span class="Number">30</span>, <span class="Number">5</span> )</span>
<span class="line"><span class="anchor" id="line-2-7"></span><span class="ID">array</span>([<span class="Number">10</span>, <span class="Number">15</span>, <span class="Number">20</span>, <span class="Number">25</span>])</span>
<span class="line"><span class="anchor" id="line-3-7"></span>>>> <span class="ID">arange</span>( <span class="Number">0</span>, <span class="Number">2</span>, <span class="Number">0.3</span> )             <span class="Comment"># it accepts float arguments</span></span>
<span class="line"><span class="anchor" id="line-4-6"></span><span class="ID">array</span>([ <span class="Number">0.</span> ,<span class="Number">0.3</span>,<span class="Number">0.6</span>,<span class="Number">0.9</span>,<span class="Number">1.2</span>,<span class="Number">1.5</span>,<span class="Number">1.8</span>])</span>
</pre></div></div><span class="anchor" id="line-141"></span><p class="line862">When <tt class="backtick">arange</tt> is used with floating point arguments, it is generally not possible to predict the number of elements obtained, due to the finite floating point precision. For this reason, it is usually better to use the function <tt class="backtick">linspace</tt> that receives as an argument the number of elements that we want, instead of the step: <span class="anchor" id="line-142"></span><span class="anchor" id="line-143"></span></p><p class="line867"><span class="anchor" id="line-144"></span><span class="anchor" id="line-145"></span><span class="anchor" id="line-146"></span><span class="anchor" id="line-147"></span><span class="anchor" id="line-148"></span><span class="anchor" id="line-149"></span><span class="anchor" id="line-1-15"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-d917b13b2e9e10887fce7dcf0fbfeed7604289b6" lang="en"><span class="line"><span class="anchor" id="line-1-16"></span>>>> <span class="ID">linspace</span>( <span class="Number">0</span>, <span class="Number">2</span>, <span class="Number">9</span> )             <span class="Comment"># 9 numbers from 0 to 2</span></span>
<span class="line"><span class="anchor" id="line-2-8"></span><span class="ID">array</span>([ <span class="Number">0.</span>,<span class="Number">0.25</span>,<span class="Number">0.5</span> ,<span class="Number">0.75</span>,<span class="Number">1.</span>,<span class="Number">1.25</span>,<span class="Number">1.5</span> ,<span class="Number">1.75</span>,<span class="Number">2.</span>])</span>
<span class="line"><span class="anchor" id="line-3-8"></span>>>> <span class="ID">x</span> = <span class="ID">linspace</span>( <span class="Number">0</span>, <span class="Number">2</span>*<span class="ID">pi</span>, <span class="Number">100</span> )    <span class="Comment"># useful to evaluate function at lots of points</span></span>
<span class="line"><span class="anchor" id="line-4-7"></span>>>> <span class="ID">f</span> = <span class="ID">sin</span>(<span class="ID">x</span>)</span>
</pre></div></div><span class="anchor" id="line-150"></span><dl><dt>See also</dt><dd><p class="line891"><a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#array">array</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#zeros">zeros</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#zeros_like">zeros_like</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#ones">ones</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#ones_like">ones_like</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#empty">empty</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#empty_like">empty_like</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#arange">arange</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#linspace">linspace</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#rand">rand</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#randn">randn</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#fromfunction">fromfunction</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#fromfile">fromfile</a> <span class="anchor" id="line-151"></span></p></dd></dl><p class="line867">
</p><h4 id="Printing_Arrays">Printing Arrays</h4>
<span class="anchor" id="line-152"></span><p class="line862">When you print an array, NumPy displays it in a similar way to nested lists, but with the following layout: <span class="anchor" id="line-153"></span><span class="anchor" id="line-154"></span></p><ul><li>the last axis is printed from left to right, <span class="anchor" id="line-155"></span></li><li>the second-to-last is printed from top to bottom, <span class="anchor" id="line-156"></span></li><li>the rest are also printed from top to bottom, with each slice separated from the next by an empty line. <span class="anchor" id="line-157"></span></li></ul><p class="line874">One-dimensional arrays are then printed as rows, bidimensionals as matrices and tridimensionals as lists of matrices. <span class="anchor" id="line-158"></span><span class="anchor" id="line-159"></span></p><p class="line867"><span class="anchor" id="line-160"></span><span class="anchor" id="line-161"></span><span class="anchor" id="line-162"></span><span class="anchor" id="line-163"></span><span class="anchor" id="line-164"></span><span class="anchor" id="line-165"></span><span class="anchor" id="line-166"></span><span class="anchor" id="line-167"></span><span class="anchor" id="line-168"></span><span class="anchor" id="line-169"></span><span class="anchor" id="line-170"></span><span class="anchor" id="line-171"></span><span class="anchor" id="line-172"></span><span class="anchor" id="line-173"></span><span class="anchor" id="line-174"></span><span class="anchor" id="line-175"></span><span class="anchor" id="line-176"></span><span class="anchor" id="line-177"></span><span class="anchor" id="line-178"></span><span class="anchor" id="line-179"></span><span class="anchor" id="line-180"></span><span class="anchor" id="line-181"></span><span class="anchor" id="line-1-17"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-5b836c3975e999e030366fe488f4013e9758616c" lang="en"><span class="line"><span class="anchor" id="line-1-18"></span>>>> <span class="ID">a</span> = <span class="ID">arange</span>(<span class="Number">6</span>)                      <span class="Comment"># 1d array</span></span>
<span class="line"><span class="anchor" id="line-2-9"></span>>>> <span class="ResWord">print</span> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-3-9"></span>[<span class="Number">0</span> <span class="Number">1</span> <span class="Number">2</span> <span class="Number">3</span> <span class="Number">4</span> <span class="Number">5</span>]</span>
<span class="line"><span class="anchor" id="line-4-8"></span>>>></span>
<span class="line"><span class="anchor" id="line-5-4"></span>>>> <span class="ID">b</span> = <span class="ID">arange</span>(<span class="Number">12</span>).<span class="ID">reshape</span>(<span class="Number">4</span>,<span class="Number">3</span>)       <span class="Comment"># 2d array</span></span>
<span class="line"><span class="anchor" id="line-6-4"></span>>>> <span class="ResWord">print</span> <span class="ID">b</span></span>
<span class="line"><span class="anchor" id="line-7-4"></span>[[ <span class="Number">0</span><span class="Number">1</span><span class="Number">2</span>]</span>
<span class="line"><span class="anchor" id="line-8-4"></span> [ <span class="Number">3</span><span class="Number">4</span><span class="Number">5</span>]</span>
<span class="line"><span class="anchor" id="line-9-4"></span> [ <span class="Number">6</span><span class="Number">7</span><span class="Number">8</span>]</span>
<span class="line"><span class="anchor" id="line-10-3"></span> [ <span class="Number">9</span> <span class="Number">10</span> <span class="Number">11</span>]]</span>
<span class="line"><span class="anchor" id="line-11-3"></span>>>></span>
<span class="line"><span class="anchor" id="line-12-3"></span>>>> <span class="ID">c</span> = <span class="ID">arange</span>(<span class="Number">24</span>).<span class="ID">reshape</span>(<span class="Number">2</span>,<span class="Number">3</span>,<span class="Number">4</span>)       <span class="Comment"># 3d array</span></span>
<span class="line"><span class="anchor" id="line-13-3"></span>>>> <span class="ResWord">print</span> <span class="ID">c</span></span>
<span class="line"><span class="anchor" id="line-14-3"></span>[[[ <span class="Number">0</span><span class="Number">1</span><span class="Number">2</span><span class="Number">3</span>]</span>
<span class="line"><span class="anchor" id="line-15-2"></span>[ <span class="Number">4</span><span class="Number">5</span><span class="Number">6</span><span class="Number">7</span>]</span>
<span class="line"><span class="anchor" id="line-16-2"></span>[ <span class="Number">8</span><span class="Number">9</span> <span class="Number">10</span> <span class="Number">11</span>]]</span>
<span class="line"><span class="anchor" id="line-17-2"></span></span>
<span class="line"><span class="anchor" id="line-18-2"></span> [[<span class="Number">12</span> <span class="Number">13</span> <span class="Number">14</span> <span class="Number">15</span>]</span>
<span class="line"><span class="anchor" id="line-19-2"></span>[<span class="Number">16</span> <span class="Number">17</span> <span class="Number">18</span> <span class="Number">19</span>]</span>
<span class="line"><span class="anchor" id="line-20-2"></span>[<span class="Number">20</span> <span class="Number">21</span> <span class="Number">22</span> <span class="Number">23</span>]]]</span>
</pre></div></div><span class="anchor" id="line-182"></span><p class="line862">See <a href="Tentative_NumPy_Tutorial.html#Shape_Manipulation">below</a> to get more details on <tt class="backtick">reshape</tt>. <span class="anchor" id="line-183"></span><span class="anchor" id="line-184"></span></p><p class="line862">If an array is too large to be printed, NumPy automatically skips the central part of the array and only prints the corners: <span class="anchor" id="line-185"></span><span class="anchor" id="line-186"></span></p><p class="line867"><span class="anchor" id="line-187"></span><span class="anchor" id="line-188"></span><span class="anchor" id="line-189"></span><span class="anchor" id="line-190"></span><span class="anchor" id="line-191"></span><span class="anchor" id="line-192"></span><span class="anchor" id="line-193"></span><span class="anchor" id="line-194"></span><span class="anchor" id="line-195"></span><span class="anchor" id="line-196"></span><span class="anchor" id="line-197"></span><span class="anchor" id="line-198"></span><span class="anchor" id="line-199"></span><span class="anchor" id="line-1-19"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-05594e79df6fbe9f9aa938422554abe2506f479a" lang="en"><span class="line"><span class="anchor" id="line-1-20"></span>>>> <span class="ResWord">print</span> <span class="ID">arange</span>(<span class="Number">10000</span>)</span>
<span class="line"><span class="anchor" id="line-2-10"></span>[ <span class="Number">0</span> <span class="Number">1</span> <span class="Number">2</span> ..., <span class="Number">9997</span> <span class="Number">9998</span> <span class="Number">9999</span>]</span>
<span class="line"><span class="anchor" id="line-3-10"></span>>>></span>
<span class="line"><span class="anchor" id="line-4-9"></span>>>> <span class="ResWord">print</span> <span class="ID">arange</span>(<span class="Number">10000</span>).<span class="ID">reshape</span>(<span class="Number">100</span>,<span class="Number">100</span>)</span>
<span class="line"><span class="anchor" id="line-5-5"></span>[[ <span class="Number">0</span> <span class="Number">1</span> <span class="Number">2</span> ..., <span class="Number">97</span> <span class="Number">98</span> <span class="Number">99</span>]</span>
<span class="line"><span class="anchor" id="line-6-5"></span> [ <span class="Number">100</span><span class="Number">101</span><span class="Number">102</span> ...,<span class="Number">197</span><span class="Number">198</span><span class="Number">199</span>]</span>
<span class="line"><span class="anchor" id="line-7-5"></span> [ <span class="Number">200</span><span class="Number">201</span><span class="Number">202</span> ...,<span class="Number">297</span><span class="Number">298</span><span class="Number">299</span>]</span>
<span class="line"><span class="anchor" id="line-8-5"></span> ...,</span>
<span class="line"><span class="anchor" id="line-9-5"></span> [<span class="Number">9700</span> <span class="Number">9701</span> <span class="Number">9702</span> ..., <span class="Number">9797</span> <span class="Number">9798</span> <span class="Number">9799</span>]</span>
<span class="line"><span class="anchor" id="line-10-4"></span> [<span class="Number">9800</span> <span class="Number">9801</span> <span class="Number">9802</span> ..., <span class="Number">9897</span> <span class="Number">9898</span> <span class="Number">9899</span>]</span>
<span class="line"><span class="anchor" id="line-11-4"></span> [<span class="Number">9900</span> <span class="Number">9901</span> <span class="Number">9902</span> ..., <span class="Number">9997</span> <span class="Number">9998</span> <span class="Number">9999</span>]]</span>
</pre></div></div><span class="anchor" id="line-200"></span><p class="line862">To disable this behaviour and force NumPy to print the entire array, you can change the printing options using <tt class="backtick">set_printoptions</tt>. <span class="anchor" id="line-201"></span><span class="anchor" id="line-202"></span></p><p class="line867"><span class="anchor" id="line-203"></span><span class="anchor" id="line-204"></span><span class="anchor" id="line-205"></span><span class="anchor" id="line-1-21"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-f7464d7b67b58758fc54a75bf847faab99b1a975" lang="en"><span class="line"><span class="anchor" id="line-1-22"></span>>>> <span class="ID">set_printoptions</span>(<span class="ID">threshold</span>=<span class="String">'</span><span class="String">nan</span><span class="String">'</span>)</span>
</pre></div></div><span class="anchor" id="line-206"></span><span class="anchor" id="line-207"></span><p class="line867">
</p><h4 id="Basic_Operations">Basic Operations</h4>
<span class="anchor" id="line-208"></span><p class="line862">Arithmetic operators on arrays apply <em>elementwise</em>. A new array is created and filled with the result. <span class="anchor" id="line-209"></span><span class="anchor" id="line-210"></span></p><p class="line867"><span class="anchor" id="line-211"></span><span class="anchor" id="line-212"></span><span class="anchor" id="line-213"></span><span class="anchor" id="line-214"></span><span class="anchor" id="line-215"></span><span class="anchor" id="line-216"></span><span class="anchor" id="line-217"></span><span class="anchor" id="line-218"></span><span class="anchor" id="line-219"></span><span class="anchor" id="line-220"></span><span class="anchor" id="line-221"></span><span class="anchor" id="line-222"></span><span class="anchor" id="line-223"></span><span class="anchor" id="line-224"></span><span class="anchor" id="line-225"></span><span class="anchor" id="line-1-23"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-209ea2175cb7932f1e62c767996e79d46780bdaa" lang="en"><span class="line"><span class="anchor" id="line-1-24"></span>>>> <span class="ID">a</span> = <span class="ID">array</span>( [<span class="Number">20</span>,<span class="Number">30</span>,<span class="Number">40</span>,<span class="Number">50</span>] )</span>
<span class="line"><span class="anchor" id="line-2-11"></span>>>> <span class="ID">b</span> = <span class="ID">arange</span>( <span class="Number">4</span> )</span>
<span class="line"><span class="anchor" id="line-3-11"></span>>>> <span class="ID">b</span></span>
<span class="line"><span class="anchor" id="line-4-10"></span><span class="ID">array</span>([<span class="Number">0</span>, <span class="Number">1</span>, <span class="Number">2</span>, <span class="Number">3</span>])</span>
<span class="line"><span class="anchor" id="line-5-6"></span>>>> <span class="ID">c</span> = <span class="ID">a</span>-<span class="ID">b</span></span>
<span class="line"><span class="anchor" id="line-6-6"></span>>>> <span class="ID">c</span></span>
<span class="line"><span class="anchor" id="line-7-6"></span><span class="ID">array</span>([<span class="Number">20</span>, <span class="Number">29</span>, <span class="Number">38</span>, <span class="Number">47</span>])</span>
<span class="line"><span class="anchor" id="line-8-6"></span>>>> <span class="ID">b</span>**<span class="Number">2</span></span>
<span class="line"><span class="anchor" id="line-9-6"></span><span class="ID">array</span>([<span class="Number">0</span>, <span class="Number">1</span>, <span class="Number">4</span>, <span class="Number">9</span>])</span>
<span class="line"><span class="anchor" id="line-10-5"></span>>>> <span class="Number">10</span>*<span class="ID">sin</span>(<span class="ID">a</span>)</span>
<span class="line"><span class="anchor" id="line-11-5"></span><span class="ID">array</span>([ <span class="Number">9.12945251</span>, -<span class="Number">9.88031624</span>,<span class="Number">7.4511316</span> , -<span class="Number">2.62374854</span>])</span>
<span class="line"><span class="anchor" id="line-12-4"></span>>>> <span class="ID">a</span><<span class="Number">35</span></span>
<span class="line"><span class="anchor" id="line-13-4"></span><span class="ID">array</span>([<span class="ResWord">True</span>, <span class="ResWord">True</span>, <span class="ResWord">False</span>, <span class="ResWord">False</span>], <span class="ID">dtype</span>=<span class="ResWord">bool</span>)</span>
</pre></div></div><span class="anchor" id="line-226"></span><span class="anchor" id="line-227"></span><p class="line862">Unlike in many matrix languages, the product operator <tt class="backtick">*</tt> operates elementwise in NumPy arrays. The matrix product can be performed using the <tt class="backtick">dot</tt> function or creating <tt class="backtick">matrix</tt> objects ( see matrix section of this tutorial ). <span class="anchor" id="line-228"></span><span class="anchor" id="line-229"></span><span class="anchor" id="line-230"></span><span class="anchor" id="line-231"></span><span class="anchor" id="line-232"></span><span class="anchor" id="line-233"></span><span class="anchor" id="line-234"></span><span class="anchor" id="line-235"></span><span class="anchor" id="line-236"></span><span class="anchor" id="line-237"></span><span class="anchor" id="line-238"></span><span class="anchor" id="line-239"></span><span class="anchor" id="line-240"></span><span class="anchor" id="line-1-25"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-1101c77bb7185afd5e74381d2389053e7c90e8b0" lang="en"><span class="line"><span class="anchor" id="line-1-26"></span>>>> <span class="ID">A</span> = <span class="ID">array</span>( [[<span class="Number">1</span>,<span class="Number">1</span>],</span>
<span class="line"><span class="anchor" id="line-2-12"></span>...          [<span class="Number">0</span>,<span class="Number">1</span>]] )</span>
<span class="line"><span class="anchor" id="line-3-12"></span>>>> <span class="ID">B</span> = <span class="ID">array</span>( [[<span class="Number">2</span>,<span class="Number">0</span>],</span>
<span class="line"><span class="anchor" id="line-4-11"></span>...          [<span class="Number">3</span>,<span class="Number">4</span>]] )</span>
<span class="line"><span class="anchor" id="line-5-7"></span>>>> <span class="ID">A</span>*<span class="ID">B</span>                      <span class="Comment"># elementwise product</span></span>
<span class="line"><span class="anchor" id="line-6-7"></span><span class="ID">array</span>([[<span class="Number">2</span>, <span class="Number">0</span>],</span>
<span class="line"><span class="anchor" id="line-7-7"></span>    [<span class="Number">0</span>, <span class="Number">4</span>]])</span>
<span class="line"><span class="anchor" id="line-8-7"></span>>>> <span class="ID">dot</span>(<span class="ID">A</span>,<span class="ID">B</span>)                <span class="Comment"># matrix product</span></span>
<span class="line"><span class="anchor" id="line-9-7"></span><span class="ID">array</span>([[<span class="Number">5</span>, <span class="Number">4</span>],</span>
<span class="line"><span class="anchor" id="line-10-6"></span>    [<span class="Number">3</span>, <span class="Number">4</span>]])</span>
</pre></div></div><span class="anchor" id="line-241"></span><span class="anchor" id="line-242"></span><p class="line862">Some operations, such as <tt class="backtick">+=</tt> and <tt class="backtick">*=</tt>, act in place to modify an existing array rather than create a new one. <span class="anchor" id="line-243"></span><span class="anchor" id="line-244"></span></p><p class="line867"><span class="anchor" id="line-245"></span><span class="anchor" id="line-246"></span><span class="anchor" id="line-247"></span><span class="anchor" id="line-248"></span><span class="anchor" id="line-249"></span><span class="anchor" id="line-250"></span><span class="anchor" id="line-251"></span><span class="anchor" id="line-252"></span><span class="anchor" id="line-253"></span><span class="anchor" id="line-254"></span><span class="anchor" id="line-255"></span><span class="anchor" id="line-256"></span><span class="anchor" id="line-257"></span><span class="anchor" id="line-258"></span><span class="anchor" id="line-259"></span><span class="anchor" id="line-260"></span><span class="anchor" id="line-1-27"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-ef87a7cdc46a70d5e072715e4bda1cd74e833f22" lang="en"><span class="line"><span class="anchor" id="line-1-28"></span>>>> <span class="ID">a</span> = <span class="ID">ones</span>((<span class="Number">2</span>,<span class="Number">3</span>), <span class="ID">dtype</span>=<span class="ResWord">int</span>)</span>
<span class="line"><span class="anchor" id="line-2-13"></span>>>> <span class="ID">b</span> = <span class="ID">random</span>.<span class="ID">random</span>((<span class="Number">2</span>,<span class="Number">3</span>))</span>
<span class="line"><span class="anchor" id="line-3-13"></span>>>> <span class="ID">a</span> *= <span class="Number">3</span></span>
<span class="line"><span class="anchor" id="line-4-12"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-5-8"></span><span class="ID">array</span>([[<span class="Number">3</span>, <span class="Number">3</span>, <span class="Number">3</span>],</span>
<span class="line"><span class="anchor" id="line-6-8"></span>    [<span class="Number">3</span>, <span class="Number">3</span>, <span class="Number">3</span>]])</span>
<span class="line"><span class="anchor" id="line-7-8"></span>>>> <span class="ID">b</span> += <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-8-8"></span>>>> <span class="ID">b</span></span>
<span class="line"><span class="anchor" id="line-9-8"></span><span class="ID">array</span>([[ <span class="Number">3.69092703</span>,<span class="Number">3.8324276</span> ,<span class="Number">3.0114541</span> ],</span>
<span class="line"><span class="anchor" id="line-10-7"></span>    [ <span class="Number">3.18679111</span>,<span class="Number">3.3039349</span> ,<span class="Number">3.37600289</span>]])</span>
<span class="line"><span class="anchor" id="line-11-6"></span>>>> <span class="ID">a</span> += <span class="ID">b</span>                               <span class="Comment"># b is converted to integer type</span></span>
<span class="line"><span class="anchor" id="line-12-5"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-13-5"></span><span class="ID">array</span>([[<span class="Number">6</span>, <span class="Number">6</span>, <span class="Number">6</span>],</span>
<span class="line"><span class="anchor" id="line-14-4"></span>    [<span class="Number">6</span>, <span class="Number">6</span>, <span class="Number">6</span>]])</span>
</pre></div></div><span class="anchor" id="line-261"></span><p class="line874">When operating with arrays of different types, the type of the resulting array corresponds to the more general or precise one (a behavior known as upcasting). <span class="anchor" id="line-262"></span><span class="anchor" id="line-263"></span></p><p class="line867"><span class="anchor" id="line-264"></span><span class="anchor" id="line-265"></span><span class="anchor" id="line-266"></span><span class="anchor" id="line-267"></span><span class="anchor" id="line-268"></span><span class="anchor" id="line-269"></span><span class="anchor" id="line-270"></span><span class="anchor" id="line-271"></span><span class="anchor" id="line-272"></span><span class="anchor" id="line-273"></span><span class="anchor" id="line-274"></span><span class="anchor" id="line-275"></span><span class="anchor" id="line-276"></span><span class="anchor" id="line-277"></span><span class="anchor" id="line-278"></span><span class="anchor" id="line-279"></span><span class="anchor" id="line-280"></span><span class="anchor" id="line-1-29"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-d9c75d93c2e130cd8964edeaab839b24b2591911" lang="en"><span class="line"><span class="anchor" id="line-1-30"></span>>>> <span class="ID">a</span> = <span class="ID">ones</span>(<span class="Number">3</span>, <span class="ID">dtype</span>=<span class="ID">int32</span>)</span>
<span class="line"><span class="anchor" id="line-2-14"></span>>>> <span class="ID">b</span> = <span class="ID">linspace</span>(<span class="Number">0</span>,<span class="ID">pi</span>,<span class="Number">3</span>)</span>
<span class="line"><span class="anchor" id="line-3-14"></span>>>> <span class="ID">b</span>.<span class="ID">dtype</span>.<span class="ID">name</span></span>
<span class="line"><span class="anchor" id="line-4-13"></span><span class="String">'</span><span class="String">float64</span><span class="String">'</span></span>
<span class="line"><span class="anchor" id="line-5-9"></span>>>> <span class="ID">c</span> = <span class="ID">a</span>+<span class="ID">b</span></span>
<span class="line"><span class="anchor" id="line-6-9"></span>>>> <span class="ID">c</span></span>
<span class="line"><span class="anchor" id="line-7-9"></span><span class="ID">array</span>([ <span class="Number">1.</span>    ,<span class="Number">2.57079633</span>,<span class="Number">4.14159265</span>])</span>
<span class="line"><span class="anchor" id="line-8-9"></span>>>> <span class="ID">c</span>.<span class="ID">dtype</span>.<span class="ID">name</span></span>
<span class="line"><span class="anchor" id="line-9-9"></span><span class="String">'</span><span class="String">float64</span><span class="String">'</span></span>
<span class="line"><span class="anchor" id="line-10-8"></span>>>> <span class="ID">d</span> = <span class="ID">exp</span>(<span class="ID">c</span>*<span class="Number">1j</span>)</span>
<span class="line"><span class="anchor" id="line-11-7"></span>>>> <span class="ID">d</span></span>
<span class="line"><span class="anchor" id="line-12-6"></span><span class="ID">array</span>([ <span class="Number">0.54030231</span>+<span class="Number">0.84147098j</span>, -<span class="Number">0.84147098</span>+<span class="Number">0.54030231j</span>,</span>
<span class="line"><span class="anchor" id="line-13-6"></span>    -<span class="Number">0.54030231</span>-<span class="Number">0.84147098j</span>])</span>
<span class="line"><span class="anchor" id="line-14-5"></span>>>> <span class="ID">d</span>.<span class="ID">dtype</span>.<span class="ID">name</span></span>
<span class="line"><span class="anchor" id="line-15-3"></span><span class="String">'</span><span class="String">complex128</span><span class="String">'</span></span>
</pre></div></div><span class="anchor" id="line-281"></span><span class="anchor" id="line-282"></span><p class="line862">Many unary operations, such as computing the sum of all the elements in the array, are implemented as methods of the <tt class="backtick">ndarray</tt> class. <span class="anchor" id="line-283"></span><span class="anchor" id="line-284"></span></p><p class="line867"><span class="anchor" id="line-285"></span><span class="anchor" id="line-286"></span><span class="anchor" id="line-287"></span><span class="anchor" id="line-288"></span><span class="anchor" id="line-289"></span><span class="anchor" id="line-290"></span><span class="anchor" id="line-291"></span><span class="anchor" id="line-292"></span><span class="anchor" id="line-293"></span><span class="anchor" id="line-294"></span><span class="anchor" id="line-295"></span><span class="anchor" id="line-296"></span><span class="anchor" id="line-1-31"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-d2a841f1fdd7f42d6506d69780709efe425b0d1e" lang="en"><span class="line"><span class="anchor" id="line-1-32"></span>>>> <span class="ID">a</span> = <span class="ID">random</span>.<span class="ID">random</span>((<span class="Number">2</span>,<span class="Number">3</span>))</span>
<span class="line"><span class="anchor" id="line-2-15"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-3-15"></span><span class="ID">array</span>([[ <span class="Number">0.6903007</span> ,<span class="Number">0.39168346</span>,<span class="Number">0.16524769</span>],</span>
<span class="line"><span class="anchor" id="line-4-14"></span>    [ <span class="Number">0.48819875</span>,<span class="Number">0.77188505</span>,<span class="Number">0.94792155</span>]])</span>
<span class="line"><span class="anchor" id="line-5-10"></span>>>> <span class="ID">a</span>.<span class="ID">sum</span>()</span>
<span class="line"><span class="anchor" id="line-6-10"></span><span class="Number">3.4552372100521485</span></span>
<span class="line"><span class="anchor" id="line-7-10"></span>>>> <span class="ID">a</span>.<span class="ID">min</span>()</span>
<span class="line"><span class="anchor" id="line-8-10"></span><span class="Number">0.16524768654743593</span></span>
<span class="line"><span class="anchor" id="line-9-10"></span>>>> <span class="ID">a</span>.<span class="ID">max</span>()</span>
<span class="line"><span class="anchor" id="line-10-9"></span><span class="Number">0.9479215542670073</span></span>
</pre></div></div><span class="anchor" id="line-297"></span><p class="line862">By default, these operations apply to the array as though it were a list of numbers, regardless of its shape. However, by specifying the <tt class="backtick">axis</tt> parameter you can apply an operation along the specified axis of an array: <span class="anchor" id="line-298"></span><span class="anchor" id="line-299"></span></p><p class="line867"><span class="anchor" id="line-300"></span><span class="anchor" id="line-301"></span><span class="anchor" id="line-302"></span><span class="anchor" id="line-303"></span><span class="anchor" id="line-304"></span><span class="anchor" id="line-305"></span><span class="anchor" id="line-306"></span><span class="anchor" id="line-307"></span><span class="anchor" id="line-308"></span><span class="anchor" id="line-309"></span><span class="anchor" id="line-310"></span><span class="anchor" id="line-311"></span><span class="anchor" id="line-312"></span><span class="anchor" id="line-313"></span><span class="anchor" id="line-314"></span><span class="anchor" id="line-315"></span><span class="anchor" id="line-316"></span><span class="anchor" id="line-317"></span><span class="anchor" id="line-1-33"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-d1f4066383163938552fc2a21893decdf72e5e2c" lang="en"><span class="line"><span class="anchor" id="line-1-34"></span>>>> <span class="ID">b</span> = <span class="ID">arange</span>(<span class="Number">12</span>).<span class="ID">reshape</span>(<span class="Number">3</span>,<span class="Number">4</span>)</span>
<span class="line"><span class="anchor" id="line-2-16"></span>>>> <span class="ID">b</span></span>
<span class="line"><span class="anchor" id="line-3-16"></span><span class="ID">array</span>([[ <span class="Number">0</span>,<span class="Number">1</span>,<span class="Number">2</span>,<span class="Number">3</span>],</span>
<span class="line"><span class="anchor" id="line-4-15"></span>    [ <span class="Number">4</span>,<span class="Number">5</span>,<span class="Number">6</span>,<span class="Number">7</span>],</span>
<span class="line"><span class="anchor" id="line-5-11"></span>    [ <span class="Number">8</span>,<span class="Number">9</span>, <span class="Number">10</span>, <span class="Number">11</span>]])</span>
<span class="line"><span class="anchor" id="line-6-11"></span>>>></span>
<span class="line"><span class="anchor" id="line-7-11"></span>>>> <span class="ID">b</span>.<span class="ID">sum</span>(<span class="ID">axis</span>=<span class="Number">0</span>)                         <span class="Comment"># sum of each column</span></span>
<span class="line"><span class="anchor" id="line-8-11"></span><span class="ID">array</span>([<span class="Number">12</span>, <span class="Number">15</span>, <span class="Number">18</span>, <span class="Number">21</span>])</span>
<span class="line"><span class="anchor" id="line-9-11"></span>>>></span>
<span class="line"><span class="anchor" id="line-10-10"></span>>>> <span class="ID">b</span>.<span class="ID">min</span>(<span class="ID">axis</span>=<span class="Number">1</span>)                         <span class="Comment"># min of each row</span></span>
<span class="line"><span class="anchor" id="line-11-8"></span><span class="ID">array</span>([<span class="Number">0</span>, <span class="Number">4</span>, <span class="Number">8</span>])</span>
<span class="line"><span class="anchor" id="line-12-7"></span>>>></span>
<span class="line"><span class="anchor" id="line-13-7"></span>>>> <span class="ID">b</span>.<span class="ID">cumsum</span>(<span class="ID">axis</span>=<span class="Number">1</span>)                      <span class="Comment"># cumulative sum along each row</span></span>
<span class="line"><span class="anchor" id="line-14-6"></span><span class="ID">array</span>([[ <span class="Number">0</span>,<span class="Number">1</span>,<span class="Number">3</span>,<span class="Number">6</span>],</span>
<span class="line"><span class="anchor" id="line-15-4"></span>    [ <span class="Number">4</span>,<span class="Number">9</span>, <span class="Number">15</span>, <span class="Number">22</span>],</span>
<span class="line"><span class="anchor" id="line-16-3"></span>    [ <span class="Number">8</span>, <span class="Number">17</span>, <span class="Number">27</span>, <span class="Number">38</span>]])</span>
</pre></div></div><span class="anchor" id="line-318"></span><span class="anchor" id="line-319"></span><p class="line867">
</p><h4 id="Universal_Functions">Universal Functions</h4>
<span class="anchor" id="line-320"></span><p class="line867">NumPy provides familiar mathematical functions such as sin, cos, and exp.In <a class="nonexistent" href="https://scipy.github.io/old-wiki/pages/NumPy.html">NumPy</a>, these are called "universal functions"(<tt class="backtick">ufunc</tt>). Within <a class="nonexistent" href="https://scipy.github.io/old-wiki/pages/NumPy.html">NumPy</a>, these functions operate elementwise on an array, producing an array as output. <span class="anchor" id="line-321"></span><span class="anchor" id="line-322"></span></p><p class="line867"><span class="anchor" id="line-323"></span><span class="anchor" id="line-324"></span><span class="anchor" id="line-325"></span><span class="anchor" id="line-326"></span><span class="anchor" id="line-327"></span><span class="anchor" id="line-328"></span><span class="anchor" id="line-329"></span><span class="anchor" id="line-330"></span><span class="anchor" id="line-331"></span><span class="anchor" id="line-332"></span><span class="anchor" id="line-333"></span><span class="anchor" id="line-334"></span><span class="anchor" id="line-1-35"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-8ab3bf6cd97fab1dfaf8aa179c73bac05b970ce0" lang="en"><span class="line"><span class="anchor" id="line-1-36"></span>>>> <span class="ID">B</span> = <span class="ID">arange</span>(<span class="Number">3</span>)</span>
<span class="line"><span class="anchor" id="line-2-17"></span>>>> <span class="ID">B</span></span>
<span class="line"><span class="anchor" id="line-3-17"></span><span class="ID">array</span>([<span class="Number">0</span>, <span class="Number">1</span>, <span class="Number">2</span>])</span>
<span class="line"><span class="anchor" id="line-4-16"></span>>>> <span class="ID">exp</span>(<span class="ID">B</span>)</span>
<span class="line"><span class="anchor" id="line-5-12"></span><span class="ID">array</span>([ <span class="Number">1.</span>    ,<span class="Number">2.71828183</span>,<span class="Number">7.3890561</span> ])</span>
<span class="line"><span class="anchor" id="line-6-12"></span>>>> <span class="ID">sqrt</span>(<span class="ID">B</span>)</span>
<span class="line"><span class="anchor" id="line-7-12"></span><span class="ID">array</span>([ <span class="Number">0.</span>    ,<span class="Number">1.</span>    ,<span class="Number">1.41421356</span>])</span>
<span class="line"><span class="anchor" id="line-8-12"></span>>>> <span class="ID">C</span> = <span class="ID">array</span>([<span class="Number">2.</span>, -<span class="Number">1.</span>, <span class="Number">4.</span>])</span>
<span class="line"><span class="anchor" id="line-9-12"></span>>>> <span class="ID">add</span>(<span class="ID">B</span>, <span class="ID">C</span>)</span>
<span class="line"><span class="anchor" id="line-10-11"></span><span class="ID">array</span>([ <span class="Number">2.</span>,<span class="Number">0.</span>,<span class="Number">6.</span>])</span>
</pre></div></div><span class="anchor" id="line-335"></span><span class="anchor" id="line-336"></span><span class="anchor" id="line-337"></span><dl><dt>See also</dt><dd><p class="line891"><a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#all">all</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#alltrue">alltrue</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#any">any</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#apply_along_axis">apply along axis</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#argmax">argmax</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#argmin">argmin</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#argsort">argsort</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#average">average</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#bincount">bincount</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#ceil">ceil</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#clip">clip</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#conj">conj</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#conjugate">conjugate</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#corrcoef">corrcoef</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#cov">cov</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#cross">cross</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#cumprod">cumprod</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#cumsum">cumsum</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#diff">diff</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#dot">dot</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#floor">floor</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#inner">inner</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#inv">inv</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#lexsort">lexsort</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#max">max</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#maximum">maximum</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#mean">mean</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#median">median</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#min">min</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#minimum">minimum</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#nonzero">nonzero</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#outer">outer</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#prod">prod</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#re">re</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#round">round</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#sometrue">sometrue</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#sort">sort</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#std">std</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#sum">sum</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#trace">trace</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#transpose">transpose</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#var">var</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#vdot">vdot</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#vectorize">vectorize</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#where">where</a> <span class="anchor" id="line-338"></span><span class="anchor" id="line-339"></span></p></dd></dl><p class="line867">
</p><h4 id="Indexing.2C_Slicing_and_Iterating">Indexing, Slicing and Iterating</h4>
<span class="anchor" id="line-340"></span><p class="line867"><strong>One-dimensional</strong> arrays can be indexed, sliced and iterated over, much like <ahref="http://docs.python.org/tut/node5.html#SECTION005140000000000000000">lists</a> and other Python sequences. <span class="anchor" id="line-341"></span><span class="anchor" id="line-342"></span></p><p class="line867"><span class="anchor" id="line-343"></span><span class="anchor" id="line-344"></span><span class="anchor" id="line-345"></span><span class="anchor" id="line-346"></span><span class="anchor" id="line-347"></span><span class="anchor" id="line-348"></span><span class="anchor" id="line-349"></span><span class="anchor" id="line-350"></span><span class="anchor" id="line-351"></span><span class="anchor" id="line-352"></span><span class="anchor" id="line-353"></span><span class="anchor" id="line-354"></span><span class="anchor" id="line-355"></span><span class="anchor" id="line-356"></span><span class="anchor" id="line-357"></span><span class="anchor" id="line-358"></span><span class="anchor" id="line-359"></span><span class="anchor" id="line-360"></span><span class="anchor" id="line-1-37"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-bbdc3dee92f44e7c89345ed71f8fafdf1bf46497" lang="en"><span class="line"><span class="anchor" id="line-1-38"></span>>>> <span class="ID">a</span> = <span class="ID">arange</span>(<span class="Number">10</span>)**<span class="Number">3</span></span>
<span class="line"><span class="anchor" id="line-2-18"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-3-18"></span><span class="ID">array</span>([<span class="Number">0</span>, <span class="Number">1</span>, <span class="Number">8</span>,<span class="Number">27</span>,<span class="Number">64</span>, <span class="Number">125</span>, <span class="Number">216</span>, <span class="Number">343</span>, <span class="Number">512</span>, <span class="Number">729</span>])</span>
<span class="line"><span class="anchor" id="line-4-17"></span>>>> <span class="ID">a</span>[<span class="Number">2</span>]</span>
<span class="line"><span class="anchor" id="line-5-13"></span><span class="Number">8</span></span>
<span class="line"><span class="anchor" id="line-6-13"></span>>>> <span class="ID">a</span>[<span class="Number">2</span>:<span class="Number">5</span>]</span>
<span class="line"><span class="anchor" id="line-7-13"></span><span class="ID">array</span>([ <span class="Number">8</span>, <span class="Number">27</span>, <span class="Number">64</span>])</span>
<span class="line"><span class="anchor" id="line-8-13"></span>>>> <span class="ID">a</span>[:<span class="Number">6</span>:<span class="Number">2</span>] = -<span class="Number">1000</span> <span class="Comment"># equivalent to a = -1000; from start to position 6, exclusive, set every 2nd element to -1000</span></span>
<span class="line"><span class="anchor" id="line-9-13"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-10-12"></span><span class="ID">array</span>([-<span class="Number">1000</span>, <span class="Number">1</span>, -<span class="Number">1000</span>, <span class="Number">27</span>, -<span class="Number">1000</span>, <span class="Number">125</span>, <span class="Number">216</span>, <span class="Number">343</span>, <span class="Number">512</span>, <span class="Number">729</span>])</span>
<span class="line"><span class="anchor" id="line-11-9"></span>>>> <span class="ID">a</span>[ : :-<span class="Number">1</span>]                               <span class="Comment"># reversed a</span></span>
<span class="line"><span class="anchor" id="line-12-8"></span><span class="ID">array</span>([<span class="Number">729</span>, <span class="Number">512</span>, <span class="Number">343</span>, <span class="Number">216</span>, <span class="Number">125</span>, -<span class="Number">1000</span>, <span class="Number">27</span>, -<span class="Number">1000</span>, <span class="Number">1</span>, -<span class="Number">1000</span>])</span>
<span class="line"><span class="anchor" id="line-13-8"></span>>>> <span class="ResWord">for</span> <span class="ID">i</span> <span class="ResWord">in</span> <span class="ID">a</span>:</span>
<span class="line"><span class="anchor" id="line-14-7"></span>...       <span class="ResWord">print</span> <span class="ID">i</span>**(<span class="Number">1</span>/<span class="Number">3.</span>),</span>
<span class="line"><span class="anchor" id="line-15-5"></span>...</span>
<span class="line"><span class="anchor" id="line-16-4"></span><span class="ID">nan</span> <span class="Number">1.0</span> <span class="ID">nan</span> <span class="Number">3.0</span> <span class="ID">nan</span> <span class="Number">5.0</span> <span class="Number">6.0</span> <span class="Number">7.0</span> <span class="Number">8.0</span> <span class="Number">9.0</span></span>
</pre></div></div><span class="anchor" id="line-361"></span><p class="line867"><strong>Multidimensional</strong> arrays can have one index per axis. These indices are given in a tuple separated by commas: <span class="anchor" id="line-362"></span><span class="anchor" id="line-363"></span></p><p class="line867"><span class="anchor" id="line-364"></span><span class="anchor" id="line-365"></span><span class="anchor" id="line-366"></span><span class="anchor" id="line-367"></span><span class="anchor" id="line-368"></span><span class="anchor" id="line-369"></span><span class="anchor" id="line-370"></span><span class="anchor" id="line-371"></span><span class="anchor" id="line-372"></span><span class="anchor" id="line-373"></span><span class="anchor" id="line-374"></span><span class="anchor" id="line-375"></span><span class="anchor" id="line-376"></span><span class="anchor" id="line-377"></span><span class="anchor" id="line-378"></span><span class="anchor" id="line-379"></span><span class="anchor" id="line-380"></span><span class="anchor" id="line-381"></span><span class="anchor" id="line-382"></span><span class="anchor" id="line-383"></span><span class="anchor" id="line-384"></span><span class="anchor" id="line-1-39"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-aaa22f7575b88784bc6244736e43d2b02ccf7a10" lang="en"><span class="line"><span class="anchor" id="line-1-40"></span>>>> <span class="ResWord">def</span> <span class="ID">f</span>(<span class="ID">x</span>,<span class="ID">y</span>):</span>
<span class="line"><span class="anchor" id="line-2-19"></span>...       <span class="ResWord">return</span> <span class="Number">10</span>*<span class="ID">x</span>+<span class="ID">y</span></span>
<span class="line"><span class="anchor" id="line-3-19"></span>...</span>
<span class="line"><span class="anchor" id="line-4-18"></span>>>> <span class="ID">b</span> = <span class="ID">fromfunction</span>(<span class="ID">f</span>,(<span class="Number">5</span>,<span class="Number">4</span>),<span class="ID">dtype</span>=<span class="ResWord">int</span>)</span>
<span class="line"><span class="anchor" id="line-5-14"></span>>>> <span class="ID">b</span></span>
<span class="line"><span class="anchor" id="line-6-14"></span><span class="ID">array</span>([[ <span class="Number">0</span>,<span class="Number">1</span>,<span class="Number">2</span>,<span class="Number">3</span>],</span>
<span class="line"><span class="anchor" id="line-7-14"></span>    [<span class="Number">10</span>, <span class="Number">11</span>, <span class="Number">12</span>, <span class="Number">13</span>],</span>
<span class="line"><span class="anchor" id="line-8-14"></span>    [<span class="Number">20</span>, <span class="Number">21</span>, <span class="Number">22</span>, <span class="Number">23</span>],</span>
<span class="line"><span class="anchor" id="line-9-14"></span>    [<span class="Number">30</span>, <span class="Number">31</span>, <span class="Number">32</span>, <span class="Number">33</span>],</span>
<span class="line"><span class="anchor" id="line-10-13"></span>    [<span class="Number">40</span>, <span class="Number">41</span>, <span class="Number">42</span>, <span class="Number">43</span>]])</span>
<span class="line"><span class="anchor" id="line-11-10"></span>>>> <span class="ID">b</span>[<span class="Number">2</span>,<span class="Number">3</span>]</span>
<span class="line"><span class="anchor" id="line-12-9"></span><span class="Number">23</span></span>
<span class="line"><span class="anchor" id="line-13-9"></span>>>> <span class="ID">b</span>[<span class="Number">0</span>:<span class="Number">5</span>, <span class="Number">1</span>]                   <span class="Comment"># each row in the second column of b</span></span>
<span class="line"><span class="anchor" id="line-14-8"></span><span class="ID">array</span>([ <span class="Number">1</span>, <span class="Number">11</span>, <span class="Number">21</span>, <span class="Number">31</span>, <span class="Number">41</span>])</span>
<span class="line"><span class="anchor" id="line-15-6"></span>>>> <span class="ID">b</span>[ : ,<span class="Number">1</span>]                      <span class="Comment"># equivalent to the previous example</span></span>
<span class="line"><span class="anchor" id="line-16-5"></span><span class="ID">array</span>([ <span class="Number">1</span>, <span class="Number">11</span>, <span class="Number">21</span>, <span class="Number">31</span>, <span class="Number">41</span>])</span>
<span class="line"><span class="anchor" id="line-17-3"></span>>>> <span class="ID">b</span>[<span class="Number">1</span>:<span class="Number">3</span>, : ]                   <span class="Comment"># each column in the second and third row of b</span></span>
<span class="line"><span class="anchor" id="line-18-3"></span><span class="ID">array</span>([[<span class="Number">10</span>, <span class="Number">11</span>, <span class="Number">12</span>, <span class="Number">13</span>],</span>
<span class="line"><span class="anchor" id="line-19-3"></span>    [<span class="Number">20</span>, <span class="Number">21</span>, <span class="Number">22</span>, <span class="Number">23</span>]])</span>
</pre></div></div><span class="anchor" id="line-385"></span><p class="line862">When fewer indices are provided than the number of axes, the missing indices are considered complete slices<tt class="backtick">:</tt> <span class="anchor" id="line-386"></span><span class="anchor" id="line-387"></span></p><p class="line867"><span class="anchor" id="line-388"></span><span class="anchor" id="line-389"></span><span class="anchor" id="line-390"></span><span class="anchor" id="line-391"></span><span class="anchor" id="line-1-41"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-f633edb5ad9be0bd3f2af6f59269c4f6af065f9c" lang="en"><span class="line"><span class="anchor" id="line-1-42"></span>>>> <span class="ID">b</span>[-<span class="Number">1</span>]                               <span class="Comment"># the last row. Equivalent to b[-1,:]</span></span>
<span class="line"><span class="anchor" id="line-2-20"></span><span class="ID">array</span>([<span class="Number">40</span>, <span class="Number">41</span>, <span class="Number">42</span>, <span class="Number">43</span>])</span>
</pre></div></div><span class="anchor" id="line-392"></span><p class="line862">The expression within brackets in <tt class="backtick">b</tt> is treated as an <tt class="backtick">i</tt> followed by as many instances of <tt class="backtick">:</tt> as needed to represent the remaining axes. NumPy also allows you to write this using dots as <tt class="backtick">b</tt>. <span class="anchor" id="line-393"></span><span class="anchor" id="line-394"></span></p><p class="line862">The <strong>dots</strong> (<tt class="backtick">...</tt>) represent as many colons as needed to produce a complete indexing tuple. For example, if <tt class="backtick">x</tt> is a rank 5 array (i.e., it has 5 axes), then <span class="anchor" id="line-395"></span><span class="anchor" id="line-396"></span></p><ul><li><p class="line891"><tt class="backtick">x</tt> is equivalent to <tt class="backtick">x</tt>, <span class="anchor" id="line-397"></span></p></li><li><p class="line891"><tt class="backtick">x[...,3]</tt> to <tt class="backtick">x[:,:,:,:,3]</tt> and <span class="anchor" id="line-398"></span></p></li><li><p class="line891"><tt class="backtick">x</tt> to <tt class="backtick">x</tt>. <span class="anchor" id="line-399"></span></p></li></ul><p class="line867"><span class="anchor" id="line-400"></span><span class="anchor" id="line-401"></span><span class="anchor" id="line-402"></span><span class="anchor" id="line-403"></span><span class="anchor" id="line-404"></span><span class="anchor" id="line-405"></span><span class="anchor" id="line-406"></span><span class="anchor" id="line-407"></span><span class="anchor" id="line-408"></span><span class="anchor" id="line-409"></span><span class="anchor" id="line-410"></span><span class="anchor" id="line-411"></span><span class="anchor" id="line-412"></span><span class="anchor" id="line-413"></span><span class="anchor" id="line-414"></span><span class="anchor" id="line-1-43"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-3d3959a08ba9199752aa62605c99e3fc94e5cd09" lang="en"><span class="line"><span class="anchor" id="line-1-44"></span>>>> <span class="ID">c</span> = <span class="ID">array</span>( [ [[<span class="Number">0</span>,<span class="Number">1</span>,<span class="Number">2</span>],             <span class="Comment"># a 3D array (two stacked 2D arrays)</span></span>
<span class="line"><span class="anchor" id="line-2-21"></span>...             [ <span class="Number">10</span>, <span class="Number">12</span>, <span class="Number">13</span>]],</span>
<span class="line"><span class="anchor" id="line-3-20"></span>...</span>
<span class="line"><span class="anchor" id="line-4-19"></span>...          [[<span class="Number">100</span>,<span class="Number">101</span>,<span class="Number">102</span>],</span>
<span class="line"><span class="anchor" id="line-5-15"></span>...             [<span class="Number">110</span>,<span class="Number">112</span>,<span class="Number">113</span>]] ] )</span>
<span class="line"><span class="anchor" id="line-6-15"></span>>>> <span class="ID">c</span>.<span class="ID">shape</span></span>
<span class="line"><span class="anchor" id="line-7-15"></span>(<span class="Number">2</span>, <span class="Number">2</span>, <span class="Number">3</span>)</span>
<span class="line"><span class="anchor" id="line-8-15"></span>>>> <span class="ID">c</span>[<span class="Number">1</span>,...]                               <span class="Comment"># same as c or c</span></span>
<span class="line"><span class="anchor" id="line-9-15"></span><span class="ID">array</span>([[<span class="Number">100</span>, <span class="Number">101</span>, <span class="Number">102</span>],</span>
<span class="line"><span class="anchor" id="line-10-14"></span>    [<span class="Number">110</span>, <span class="Number">112</span>, <span class="Number">113</span>]])</span>
<span class="line"><span class="anchor" id="line-11-11"></span>>>> <span class="ID">c</span>[...,<span class="Number">2</span>]                               <span class="Comment"># same as c[:,:,2]</span></span>
<span class="line"><span class="anchor" id="line-12-10"></span><span class="ID">array</span>([[<span class="Number">2</span>,<span class="Number">13</span>],</span>
<span class="line"><span class="anchor" id="line-13-10"></span>    [<span class="Number">102</span>, <span class="Number">113</span>]])</span>
</pre></div></div><span class="anchor" id="line-415"></span><p class="line867"><strong>Iterating</strong> over multidimensional arrays is done with respect to the first axis: <span class="anchor" id="line-416"></span><span class="anchor" id="line-417"></span></p><p class="line867"><span class="anchor" id="line-418"></span><span class="anchor" id="line-419"></span><span class="anchor" id="line-420"></span><span class="anchor" id="line-421"></span><span class="anchor" id="line-422"></span><span class="anchor" id="line-423"></span><span class="anchor" id="line-424"></span><span class="anchor" id="line-425"></span><span class="anchor" id="line-426"></span><span class="anchor" id="line-427"></span><span class="anchor" id="line-1-45"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-6adc57b75e45bed1bdfb71513f4878232164ef8a" lang="en"><span class="line"><span class="anchor" id="line-1-46"></span>>>> <span class="ResWord">for</span> <span class="ID">row</span> <span class="ResWord">in</span> <span class="ID">b</span>:</span>
<span class="line"><span class="anchor" id="line-2-22"></span>...       <span class="ResWord">print</span> <span class="ID">row</span></span>
<span class="line"><span class="anchor" id="line-3-21"></span>...</span>
<span class="line"><span class="anchor" id="line-4-20"></span>[<span class="Number">0</span> <span class="Number">1</span> <span class="Number">2</span> <span class="Number">3</span>]</span>
<span class="line"><span class="anchor" id="line-5-16"></span>[<span class="Number">10</span> <span class="Number">11</span> <span class="Number">12</span> <span class="Number">13</span>]</span>
<span class="line"><span class="anchor" id="line-6-16"></span>[<span class="Number">20</span> <span class="Number">21</span> <span class="Number">22</span> <span class="Number">23</span>]</span>
<span class="line"><span class="anchor" id="line-7-16"></span>[<span class="Number">30</span> <span class="Number">31</span> <span class="Number">32</span> <span class="Number">33</span>]</span>
<span class="line"><span class="anchor" id="line-8-16"></span>[<span class="Number">40</span> <span class="Number">41</span> <span class="Number">42</span> <span class="Number">43</span>]</span>
</pre></div></div><span class="anchor" id="line-428"></span><p class="line862">However, if one wants to perform an operation on each element in the array, one can use the <tt class="backtick">flat</tt> attribute which is an <ahref="http://docs.python.org/tut/node11.html#SECTION0011900000000000000000">iterator</a> over all the elements of the array: <span class="anchor" id="line-429"></span><span class="anchor" id="line-430"></span></p><p class="line867"><span class="anchor" id="line-431"></span><span class="anchor" id="line-432"></span><span class="anchor" id="line-433"></span><span class="anchor" id="line-434"></span><span class="anchor" id="line-435"></span><span class="anchor" id="line-436"></span><span class="anchor" id="line-1-47"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-ebe10519055da04e76f58eee5712ea2a62bf15f3" lang="en"><span class="line"><span class="anchor" id="line-1-48"></span>>>> <span class="ResWord">for</span> <span class="ID">element</span> <span class="ResWord">in</span> <span class="ID">b</span>.<span class="ID">flat</span>:</span>
<span class="line"><span class="anchor" id="line-2-23"></span>...       <span class="ResWord">print</span> <span class="ID">element</span>,</span>
<span class="line"><span class="anchor" id="line-3-22"></span>...</span>
<span class="line"><span class="anchor" id="line-4-21"></span><span class="Number">0</span> <span class="Number">1</span> <span class="Number">2</span> <span class="Number">3</span> <span class="Number">10</span> <span class="Number">11</span> <span class="Number">12</span> <span class="Number">13</span> <span class="Number">20</span> <span class="Number">21</span> <span class="Number">22</span> <span class="Number">23</span> <span class="Number">30</span> <span class="Number">31</span> <span class="Number">32</span> <span class="Number">33</span> <span class="Number">40</span> <span class="Number">41</span> <span class="Number">42</span> <span class="Number">43</span></span>
</pre></div></div><span class="anchor" id="line-437"></span><dl><dt>See also</dt><dd><p class="line891"><a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#bbrackets">[</a>], <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#dots">...</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#newaxis">newaxis</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#ndenumerate">ndenumerate</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#indices">indices</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#index_exp">index exp</a> <span class="anchor" id="line-438"></span></p></dd></dl><p class="line867"><span class="anchor" id="Shape_Manipulation"></span> <span class="anchor" id="line-439"></span><span class="anchor" id="line-440"></span></p><p class="line867">
</p><h2 id="Shape_Manipulation-1">Shape Manipulation</h2>
<span class="anchor" id="line-441"></span><p class="line867">
</p><h4 id="Changing_the_shape_of_an_array">Changing the shape of an array</h4>
<span class="anchor" id="line-442"></span><p class="line874">An array has a shape given by the number of elements along each axis: <span class="anchor" id="line-443"></span><span class="anchor" id="line-444"></span></p><p class="line867"><span class="anchor" id="line-445"></span><span class="anchor" id="line-446"></span><span class="anchor" id="line-447"></span><span class="anchor" id="line-448"></span><span class="anchor" id="line-449"></span><span class="anchor" id="line-450"></span><span class="anchor" id="line-451"></span><span class="anchor" id="line-452"></span><span class="anchor" id="line-453"></span><span class="anchor" id="line-1-49"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-48aaec7ec88cdc6cd3a66c59444b38ff7cdd718a" lang="en"><span class="line"><span class="anchor" id="line-1-50"></span>>>> <span class="ID">a</span> = <span class="ID">floor</span>(<span class="Number">10</span>*<span class="ID">random</span>.<span class="ID">random</span>((<span class="Number">3</span>,<span class="Number">4</span>)))</span>
<span class="line"><span class="anchor" id="line-2-24"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-3-23"></span><span class="ID">array</span>([[ <span class="Number">7.</span>,<span class="Number">5.</span>,<span class="Number">9.</span>,<span class="Number">3.</span>],</span>
<span class="line"><span class="anchor" id="line-4-22"></span>    [ <span class="Number">7.</span>,<span class="Number">2.</span>,<span class="Number">7.</span>,<span class="Number">8.</span>],</span>
<span class="line"><span class="anchor" id="line-5-17"></span>    [ <span class="Number">6.</span>,<span class="Number">8.</span>,<span class="Number">3.</span>,<span class="Number">2.</span>]])</span>
<span class="line"><span class="anchor" id="line-6-17"></span>>>> <span class="ID">a</span>.<span class="ID">shape</span></span>
<span class="line"><span class="anchor" id="line-7-17"></span>(<span class="Number">3</span>, <span class="Number">4</span>)</span>
</pre></div></div><span class="anchor" id="line-454"></span><p class="line874">The shape of an array can be changed with various commands: <span class="anchor" id="line-455"></span><span class="anchor" id="line-456"></span></p><p class="line867"><span class="anchor" id="line-457"></span><span class="anchor" id="line-458"></span><span class="anchor" id="line-459"></span><span class="anchor" id="line-460"></span><span class="anchor" id="line-461"></span><span class="anchor" id="line-462"></span><span class="anchor" id="line-463"></span><span class="anchor" id="line-464"></span><span class="anchor" id="line-1-51"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-0153e1da46850898361978ba4e1a84289bab64df" lang="en"><span class="line"><span class="anchor" id="line-1-52"></span>>>> <span class="ID">a</span>.<span class="ID">ravel</span>() <span class="Comment"># flatten the array</span></span>
<span class="line"><span class="anchor" id="line-2-25"></span><span class="ID">array</span>([ <span class="Number">7.</span>,<span class="Number">5.</span>,<span class="Number">9.</span>,<span class="Number">3.</span>,<span class="Number">7.</span>,<span class="Number">2.</span>,<span class="Number">7.</span>,<span class="Number">8.</span>,<span class="Number">6.</span>,<span class="Number">8.</span>,<span class="Number">3.</span>,<span class="Number">2.</span>])</span>
<span class="line"><span class="anchor" id="line-3-24"></span>>>> <span class="ID">a</span>.<span class="ID">shape</span> = (<span class="Number">6</span>, <span class="Number">2</span>)</span>
<span class="line"><span class="anchor" id="line-4-23"></span>>>> <span class="ID">a</span>.<span class="ID">transpose</span>()</span>
<span class="line"><span class="anchor" id="line-5-18"></span><span class="ID">array</span>([[ <span class="Number">7.</span>,<span class="Number">9.</span>,<span class="Number">7.</span>,<span class="Number">7.</span>,<span class="Number">6.</span>,<span class="Number">3.</span>],</span>
<span class="line"><span class="anchor" id="line-6-18"></span>    [ <span class="Number">5.</span>,<span class="Number">3.</span>,<span class="Number">2.</span>,<span class="Number">8.</span>,<span class="Number">8.</span>,<span class="Number">2.</span>]])</span>
</pre></div></div><span class="anchor" id="line-465"></span><p class="line874">The order of the elements in the array resulting from ravel() is normally "C-style", that is, the rightmost index "changes the fastest", so the element after a is a. If the array is reshaped to some other shape, again the array is treated as "C-style". Numpy normally creates arrays stored in this order, so ravel() will usually not need to copy its argument, but if the array was made by taking slices of another array or created with unusual options, it may need to be copied. The functions ravel() and reshape() can also be instructed, using an optional argument, to use FORTRAN-style arrays, in which the leftmost index changes the fastest. <span class="anchor" id="line-466"></span><span class="anchor" id="line-467"></span></p><p class="line862">The <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#reshape">reshape</a> function returns its argument with a modified shape, whereas the <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#resize">resize</a> method modifies the array itself: <span class="anchor" id="line-468"></span><span class="anchor" id="line-469"></span></p><p class="line867"><span class="anchor" id="line-470"></span><span class="anchor" id="line-471"></span><span class="anchor" id="line-472"></span><span class="anchor" id="line-473"></span><span class="anchor" id="line-474"></span><span class="anchor" id="line-475"></span><span class="anchor" id="line-476"></span><span class="anchor" id="line-477"></span><span class="anchor" id="line-478"></span><span class="anchor" id="line-479"></span><span class="anchor" id="line-480"></span><span class="anchor" id="line-481"></span><span class="anchor" id="line-482"></span><span class="anchor" id="line-1-53"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-86f62cfb97c57a9ae1d1f7f83e6e84538ae9ccae" lang="en"><span class="line"><span class="anchor" id="line-1-54"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-2-26"></span><span class="ID">array</span>([[ <span class="Number">7.</span>,<span class="Number">5.</span>],</span>
<span class="line"><span class="anchor" id="line-3-25"></span>    [ <span class="Number">9.</span>,<span class="Number">3.</span>],</span>
<span class="line"><span class="anchor" id="line-4-24"></span>    [ <span class="Number">7.</span>,<span class="Number">2.</span>],</span>
<span class="line"><span class="anchor" id="line-5-19"></span>    [ <span class="Number">7.</span>,<span class="Number">8.</span>],</span>
<span class="line"><span class="anchor" id="line-6-19"></span>    [ <span class="Number">6.</span>,<span class="Number">8.</span>],</span>
<span class="line"><span class="anchor" id="line-7-18"></span>    [ <span class="Number">3.</span>,<span class="Number">2.</span>]])</span>
<span class="line"><span class="anchor" id="line-8-17"></span>>>> <span class="ID">a</span>.<span class="ID">resize</span>((<span class="Number">2</span>,<span class="Number">6</span>))</span>
<span class="line"><span class="anchor" id="line-9-16"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-10-15"></span><span class="ID">array</span>([[ <span class="Number">7.</span>,<span class="Number">5.</span>,<span class="Number">9.</span>,<span class="Number">3.</span>,<span class="Number">7.</span>,<span class="Number">2.</span>],</span>
<span class="line"><span class="anchor" id="line-11-12"></span>    [ <span class="Number">7.</span>,<span class="Number">8.</span>,<span class="Number">6.</span>,<span class="Number">8.</span>,<span class="Number">3.</span>,<span class="Number">2.</span>]])</span>
</pre></div></div><span class="anchor" id="line-483"></span><span class="anchor" id="line-484"></span><p class="line874">If a dimension is given as -1 in a reshaping operation, the other dimensions are automatically calculated: <span class="anchor" id="line-485"></span><span class="anchor" id="line-486"></span></p><p class="line867"><span class="anchor" id="line-487"></span><span class="anchor" id="line-488"></span><span class="anchor" id="line-489"></span><span class="anchor" id="line-490"></span><span class="anchor" id="line-491"></span><span class="anchor" id="line-492"></span><span class="anchor" id="line-1-55"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-1905955eca4b3a4b840eaaaca5477901526db96f" lang="en"><span class="line"><span class="anchor" id="line-1-56"></span>>>> <span class="ID">a</span>.<span class="ID">reshape</span>(<span class="Number">3</span>,-<span class="Number">1</span>)</span>
<span class="line"><span class="anchor" id="line-2-27"></span><span class="ID">array</span>([[ <span class="Number">7.</span>,<span class="Number">5.</span>,<span class="Number">9.</span>,<span class="Number">3.</span>],</span>
<span class="line"><span class="anchor" id="line-3-26"></span>    [ <span class="Number">7.</span>,<span class="Number">2.</span>,<span class="Number">7.</span>,<span class="Number">8.</span>],</span>
<span class="line"><span class="anchor" id="line-4-25"></span>    [ <span class="Number">6.</span>,<span class="Number">8.</span>,<span class="Number">3.</span>,<span class="Number">2.</span>]])</span>
</pre></div></div><span class="anchor" id="line-493"></span><p class="line862">See also:: <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#shape">shape example</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#reshape">reshape example</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#resize">resize example</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#ravel">ravel example</a> <span class="anchor" id="line-494"></span><span class="anchor" id="line-495"></span></p><p class="line867">
</p><h4 id="Stacking_together_different_arrays">Stacking together different arrays</h4>
<span class="anchor" id="line-496"></span><p class="line874">Several arrays can be stacked together along different axes: <span class="anchor" id="line-497"></span><span class="anchor" id="line-498"></span></p><p class="line867"><span class="anchor" id="line-499"></span><span class="anchor" id="line-500"></span><span class="anchor" id="line-501"></span><span class="anchor" id="line-502"></span><span class="anchor" id="line-503"></span><span class="anchor" id="line-504"></span><span class="anchor" id="line-505"></span><span class="anchor" id="line-506"></span><span class="anchor" id="line-507"></span><span class="anchor" id="line-508"></span><span class="anchor" id="line-509"></span><span class="anchor" id="line-510"></span><span class="anchor" id="line-511"></span><span class="anchor" id="line-512"></span><span class="anchor" id="line-513"></span><span class="anchor" id="line-514"></span><span class="anchor" id="line-515"></span><span class="anchor" id="line-516"></span><span class="anchor" id="line-1-57"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-4a294d5a47647f034b66021354c5a624898854dd" lang="en"><span class="line"><span class="anchor" id="line-1-58"></span>>>> <span class="ID">a</span> = <span class="ID">floor</span>(<span class="Number">10</span>*<span class="ID">random</span>.<span class="ID">random</span>((<span class="Number">2</span>,<span class="Number">2</span>)))</span>
<span class="line"><span class="anchor" id="line-2-28"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-3-27"></span><span class="ID">array</span>([[ <span class="Number">1.</span>,<span class="Number">1.</span>],</span>
<span class="line"><span class="anchor" id="line-4-26"></span>    [ <span class="Number">5.</span>,<span class="Number">8.</span>]])</span>
<span class="line"><span class="anchor" id="line-5-20"></span>>>> <span class="ID">b</span> = <span class="ID">floor</span>(<span class="Number">10</span>*<span class="ID">random</span>.<span class="ID">random</span>((<span class="Number">2</span>,<span class="Number">2</span>)))</span>
<span class="line"><span class="anchor" id="line-6-20"></span>>>> <span class="ID">b</span></span>
<span class="line"><span class="anchor" id="line-7-19"></span><span class="ID">array</span>([[ <span class="Number">3.</span>,<span class="Number">3.</span>],</span>
<span class="line"><span class="anchor" id="line-8-18"></span>    [ <span class="Number">6.</span>,<span class="Number">0.</span>]])</span>
<span class="line"><span class="anchor" id="line-9-17"></span>>>> <span class="ID">vstack</span>((<span class="ID">a</span>,<span class="ID">b</span>))</span>
<span class="line"><span class="anchor" id="line-10-16"></span><span class="ID">array</span>([[ <span class="Number">1.</span>,<span class="Number">1.</span>],</span>
<span class="line"><span class="anchor" id="line-11-13"></span>    [ <span class="Number">5.</span>,<span class="Number">8.</span>],</span>
<span class="line"><span class="anchor" id="line-12-11"></span>    [ <span class="Number">3.</span>,<span class="Number">3.</span>],</span>
<span class="line"><span class="anchor" id="line-13-11"></span>    [ <span class="Number">6.</span>,<span class="Number">0.</span>]])</span>
<span class="line"><span class="anchor" id="line-14-9"></span>>>> <span class="ID">hstack</span>((<span class="ID">a</span>,<span class="ID">b</span>))</span>
<span class="line"><span class="anchor" id="line-15-7"></span><span class="ID">array</span>([[ <span class="Number">1.</span>,<span class="Number">1.</span>,<span class="Number">3.</span>,<span class="Number">3.</span>],</span>
<span class="line"><span class="anchor" id="line-16-6"></span>    [ <span class="Number">5.</span>,<span class="Number">8.</span>,<span class="Number">6.</span>,<span class="Number">0.</span>]])</span>
</pre></div></div><span class="anchor" id="line-517"></span><p class="line862">The function <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#column_stack">column_stack</a> stacks 1D arrays as columns into a 2D array. It is equivalent to <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#vstack">vstack</a> only for 1D arrays: <span class="anchor" id="line-518"></span><span class="anchor" id="line-519"></span></p><p class="line867"><span class="anchor" id="line-520"></span><span class="anchor" id="line-521"></span><span class="anchor" id="line-522"></span><span class="anchor" id="line-523"></span><span class="anchor" id="line-524"></span><span class="anchor" id="line-525"></span><span class="anchor" id="line-526"></span><span class="anchor" id="line-527"></span><span class="anchor" id="line-528"></span><span class="anchor" id="line-529"></span><span class="anchor" id="line-530"></span><span class="anchor" id="line-531"></span><span class="anchor" id="line-532"></span><span class="anchor" id="line-533"></span><span class="anchor" id="line-534"></span><span class="anchor" id="line-535"></span><span class="anchor" id="line-536"></span><span class="anchor" id="line-537"></span><span class="anchor" id="line-1-59"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-84fa6a737f2754041752ca67ec2798514b290297" lang="en"><span class="line"><span class="anchor" id="line-1-60"></span>>>> <span class="ID">column_stack</span>((<span class="ID">a</span>,<span class="ID">b</span>)) <span class="Comment"># With 2D arrays</span></span>
<span class="line"><span class="anchor" id="line-2-29"></span><span class="ID">array</span>([[ <span class="Number">1.</span>,<span class="Number">1.</span>,<span class="Number">3.</span>,<span class="Number">3.</span>],</span>
<span class="line"><span class="anchor" id="line-3-28"></span>    [ <span class="Number">5.</span>,<span class="Number">8.</span>,<span class="Number">6.</span>,<span class="Number">0.</span>]])</span>
<span class="line"><span class="anchor" id="line-4-27"></span>>>> <span class="ID">a</span>=<span class="ID">array</span>([<span class="Number">4.</span>,<span class="Number">2.</span>])</span>
<span class="line"><span class="anchor" id="line-5-21"></span>>>> <span class="ID">b</span>=<span class="ID">array</span>([<span class="Number">2.</span>,<span class="Number">8.</span>])</span>
<span class="line"><span class="anchor" id="line-6-21"></span>>>> <span class="ID">a</span>[:,<span class="ID">newaxis</span>]<span class="Comment"># This allows to have a 2D columns vector</span></span>
<span class="line"><span class="anchor" id="line-7-20"></span><span class="ID">array</span>([[ <span class="Number">4.</span>],</span>
<span class="line"><span class="anchor" id="line-8-19"></span>    [ <span class="Number">2.</span>]])</span>
<span class="line"><span class="anchor" id="line-9-18"></span>>>> <span class="ID">column_stack</span>((<span class="ID">a</span>[:,<span class="ID">newaxis</span>],<span class="ID">b</span>[:,<span class="ID">newaxis</span>]))</span>
<span class="line"><span class="anchor" id="line-10-17"></span><span class="ID">array</span>([[ <span class="Number">4.</span>,<span class="Number">2.</span>],</span>
<span class="line"><span class="anchor" id="line-11-14"></span>    [ <span class="Number">2.</span>,<span class="Number">8.</span>]])</span>
<span class="line"><span class="anchor" id="line-12-12"></span>>>> <span class="ID">vstack</span>((<span class="ID">a</span>[:,<span class="ID">newaxis</span>],<span class="ID">b</span>[:,<span class="ID">newaxis</span>])) <span class="Comment"># The behavior of vstack is different</span></span>
<span class="line"><span class="anchor" id="line-13-12"></span><span class="ID">array</span>([[ <span class="Number">4.</span>],</span>
<span class="line"><span class="anchor" id="line-14-10"></span>    [ <span class="Number">2.</span>],</span>
<span class="line"><span class="anchor" id="line-15-8"></span>    [ <span class="Number">2.</span>],</span>
<span class="line"><span class="anchor" id="line-16-7"></span>    [ <span class="Number">8.</span>]])</span>
</pre></div></div><span class="anchor" id="line-538"></span><p class="line862">The function <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#row_stack">row_stack</a>, on the other hand, stacks 1D arrays as rows into a 2D array. <span class="anchor" id="line-539"></span><span class="anchor" id="line-540"></span></p><p class="line862">For arrays of with more than two dimensions, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#hstack">hstack</a> stacks along their second axes, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#vstack">vstack</a> stacks along their first axes, and <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#concatenate">concatenate</a> allows for an optional arguments giving the number of the axis along which the concatenation should happen. <span class="anchor" id="line-541"></span><span class="anchor" id="line-542"></span></p><p class="line867"><strong>Note</strong> <span class="anchor" id="line-543"></span><span class="anchor" id="line-544"></span></p><p class="line862">In complex cases, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#r_">r_</a>[] and <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#c_">c_</a>[] are useful for creating arrays by stacking numbers along one axis. They allow the use of range literals (":") : <span class="anchor" id="line-545"></span><span class="anchor" id="line-546"></span></p><ul><li style="list-style-type:none"><span class="anchor" id="line-547"></span><span class="anchor" id="line-548"></span><span class="anchor" id="line-549"></span><span class="anchor" id="line-550"></span><span class="anchor" id="line-1-61"></span><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-be7abca8774389460f79ce71489968ab13c48b08" lang="en"><span class="line"><span class="anchor" id="line-1-62"></span>>>> <span class="ID">r_</span>[<span class="Number">1</span>:<span class="Number">4</span>,<span class="Number">0</span>,<span class="Number">4</span>]</span>
<span class="line"><span class="anchor" id="line-2-30"></span><span class="ID">array</span>([<span class="Number">1</span>, <span class="Number">2</span>, <span class="Number">3</span>, <span class="Number">0</span>, <span class="Number">4</span>])</span>
</pre></div></div><span class="anchor" id="line-551"></span></li></ul><p class="line862">When used with arrays as arguments, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#r_">r_</a>[] and <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#c_">c_</a>[] are similar to <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#vstack">vstack</a> and <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#hstack">hstack</a> in their default behavior, but allow for an optional argument giving the number of the axis along which to concatenate. <span class="anchor" id="line-552"></span><span class="anchor" id="line-553"></span></p><p class="line862">See also: <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#hstack">hstack example</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#vstack">vstack exammple</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#column_stack">column_stack example</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#row_stack">row_stack example</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#concatenate">concatenate example</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#c_">c_ example</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#r_">r_ example</a> <span class="anchor" id="line-554"></span><span class="anchor" id="line-555"></span></p><p class="line867">
</p><h4 id="Splitting_one_array_into_several_smaller_ones">Splitting one array into several smaller ones</h4>
<span class="anchor" id="line-556"></span><p class="line862">Using <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#hsplit">hsplit</a>, you can split an array along its horizontal axis, either by specifying the number of equally shaped arrays to return, or by specifying the columns after which the division should occur: <span class="anchor" id="line-557"></span><span class="anchor" id="line-558"></span></p><p class="line867"><span class="anchor" id="line-559"></span><span class="anchor" id="line-560"></span><span class="anchor" id="line-561"></span><span class="anchor" id="line-562"></span><span class="anchor" id="line-563"></span><span class="anchor" id="line-564"></span><span class="anchor" id="line-565"></span><span class="anchor" id="line-566"></span><span class="anchor" id="line-567"></span><span class="anchor" id="line-568"></span><span class="anchor" id="line-569"></span><span class="anchor" id="line-570"></span><span class="anchor" id="line-571"></span><span class="anchor" id="line-572"></span><span class="anchor" id="line-573"></span><span class="anchor" id="line-574"></span><span class="anchor" id="line-1-63"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-d5df2353407a123b9cec09a59c464edc97f33e08" lang="en"><span class="line"><span class="anchor" id="line-1-64"></span>>>> <span class="ID">a</span> = <span class="ID">floor</span>(<span class="Number">10</span>*<span class="ID">random</span>.<span class="ID">random</span>((<span class="Number">2</span>,<span class="Number">12</span>)))</span>
<span class="line"><span class="anchor" id="line-2-31"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-3-29"></span><span class="ID">array</span>([[ <span class="Number">8.</span>,<span class="Number">8.</span>,<span class="Number">3.</span>,<span class="Number">9.</span>,<span class="Number">0.</span>,<span class="Number">4.</span>,<span class="Number">3.</span>,<span class="Number">0.</span>,<span class="Number">0.</span>,<span class="Number">6.</span>,<span class="Number">4.</span>,<span class="Number">4.</span>],</span>
<span class="line"><span class="anchor" id="line-4-28"></span>    [ <span class="Number">0.</span>,<span class="Number">3.</span>,<span class="Number">2.</span>,<span class="Number">9.</span>,<span class="Number">6.</span>,<span class="Number">0.</span>,<span class="Number">4.</span>,<span class="Number">5.</span>,<span class="Number">7.</span>,<span class="Number">5.</span>,<span class="Number">1.</span>,<span class="Number">4.</span>]])</span>
<span class="line"><span class="anchor" id="line-5-22"></span>>>> <span class="ID">hsplit</span>(<span class="ID">a</span>,<span class="Number">3</span>) <span class="Comment"># Split a into 3</span></span>
<span class="line"><span class="anchor" id="line-6-22"></span>[<span class="ID">array</span>([[ <span class="Number">8.</span>,<span class="Number">8.</span>,<span class="Number">3.</span>,<span class="Number">9.</span>],</span>
<span class="line"><span class="anchor" id="line-7-21"></span>    [ <span class="Number">0.</span>,<span class="Number">3.</span>,<span class="Number">2.</span>,<span class="Number">9.</span>]]), <span class="ID">array</span>([[ <span class="Number">0.</span>,<span class="Number">4.</span>,<span class="Number">3.</span>,<span class="Number">0.</span>],</span>
<span class="line"><span class="anchor" id="line-8-20"></span>    [ <span class="Number">6.</span>,<span class="Number">0.</span>,<span class="Number">4.</span>,<span class="Number">5.</span>]]), <span class="ID">array</span>([[ <span class="Number">0.</span>,<span class="Number">6.</span>,<span class="Number">4.</span>,<span class="Number">4.</span>],</span>
<span class="line"><span class="anchor" id="line-9-19"></span>    [ <span class="Number">7.</span>,<span class="Number">5.</span>,<span class="Number">1.</span>,<span class="Number">4.</span>]])]</span>
<span class="line"><span class="anchor" id="line-10-18"></span>>>> <span class="ID">hsplit</span>(<span class="ID">a</span>,(<span class="Number">3</span>,<span class="Number">4</span>)) <span class="Comment"># Split a after the third and the fourth column</span></span>
<span class="line"><span class="anchor" id="line-11-15"></span>[<span class="ID">array</span>([[ <span class="Number">8.</span>,<span class="Number">8.</span>,<span class="Number">3.</span>],</span>
<span class="line"><span class="anchor" id="line-12-13"></span>    [ <span class="Number">0.</span>,<span class="Number">3.</span>,<span class="Number">2.</span>]]), <span class="ID">array</span>([[ <span class="Number">9.</span>],</span>
<span class="line"><span class="anchor" id="line-13-13"></span>    [ <span class="Number">9.</span>]]), <span class="ID">array</span>([[ <span class="Number">0.</span>,<span class="Number">4.</span>,<span class="Number">3.</span>,<span class="Number">0.</span>,<span class="Number">0.</span>,<span class="Number">6.</span>,<span class="Number">4.</span>,<span class="Number">4.</span>],</span>
<span class="line"><span class="anchor" id="line-14-11"></span>    [ <span class="Number">6.</span>,<span class="Number">0.</span>,<span class="Number">4.</span>,<span class="Number">5.</span>,<span class="Number">7.</span>,<span class="Number">5.</span>,<span class="Number">1.</span>,<span class="Number">4.</span>]])]</span>
</pre></div></div><span class="anchor" id="line-575"></span><p class="line867"><a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#vsplit">vsplit</a> splits along the vertical axis, and <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#array_split">array split</a> allows one to specify along which axis to split. <span class="anchor" id="line-576"></span><span class="anchor" id="line-577"></span></p><p class="line867">
</p><h2 id="Copies_and_Views">Copies and Views</h2>
<span class="anchor" id="line-578"></span><p class="line874">When operating and manipulating arrays, their data is sometimes copied into a new array and sometimes not. This is often a source of confusion for beginners. There are three cases: <span class="anchor" id="line-579"></span><span class="anchor" id="line-580"></span></p><p class="line867">
</p><h4 id="No_Copy_at_All">No Copy at All</h4>
<span class="anchor" id="line-581"></span><p class="line874">Simple assignments make no copy of array objects or of their data. <span class="anchor" id="line-582"></span><span class="anchor" id="line-583"></span></p><p class="line867"><span class="anchor" id="line-584"></span><span class="anchor" id="line-585"></span><span class="anchor" id="line-586"></span><span class="anchor" id="line-587"></span><span class="anchor" id="line-588"></span><span class="anchor" id="line-589"></span><span class="anchor" id="line-590"></span><span class="anchor" id="line-591"></span><span class="anchor" id="line-592"></span><span class="anchor" id="line-1-65"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-8134d228ff0b840390de3d9751d538164eb985a3" lang="en"><span class="line"><span class="anchor" id="line-1-66"></span>>>> <span class="ID">a</span> = <span class="ID">arange</span>(<span class="Number">12</span>)</span>
<span class="line"><span class="anchor" id="line-2-32"></span>>>> <span class="ID">b</span> = <span class="ID">a</span>          <span class="Comment"># no new object is created</span></span>
<span class="line"><span class="anchor" id="line-3-30"></span>>>> <span class="ID">b</span> <span class="ResWord">is</span> <span class="ID">a</span>       <span class="Comment"># a and b are two names for the same ndarray object</span></span>
<span class="line"><span class="anchor" id="line-4-29"></span><span class="ResWord">True</span></span>
<span class="line"><span class="anchor" id="line-5-23"></span>>>> <span class="ID">b</span>.<span class="ID">shape</span> = <span class="Number">3</span>,<span class="Number">4</span> <span class="Comment"># changes the shape of a</span></span>
<span class="line"><span class="anchor" id="line-6-23"></span>>>> <span class="ID">a</span>.<span class="ID">shape</span></span>
<span class="line"><span class="anchor" id="line-7-22"></span>(<span class="Number">3</span>, <span class="Number">4</span>)</span>
</pre></div></div><span class="anchor" id="line-593"></span><p class="line874">Python passes mutable objects as references, so function calls make no copy. <span class="anchor" id="line-594"></span><span class="anchor" id="line-595"></span></p><p class="line867"><span class="anchor" id="line-596"></span><span class="anchor" id="line-597"></span><span class="anchor" id="line-598"></span><span class="anchor" id="line-599"></span><span class="anchor" id="line-600"></span><span class="anchor" id="line-601"></span><span class="anchor" id="line-602"></span><span class="anchor" id="line-603"></span><span class="anchor" id="line-604"></span><span class="anchor" id="line-1-67"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-30c5d93670bfa4594ccd7e18e9d00bf0462aad59" lang="en"><span class="line"><span class="anchor" id="line-1-68"></span>>>> <span class="ResWord">def</span> <span class="ID">f</span>(<span class="ID">x</span>):</span>
<span class="line"><span class="anchor" id="line-2-33"></span>... <span class="ResWord">print</span> <span class="ResWord">id</span>(<span class="ID">x</span>)</span>
<span class="line"><span class="anchor" id="line-3-31"></span>...</span>
<span class="line"><span class="anchor" id="line-4-30"></span>>>> <span class="ResWord">id</span>(<span class="ID">a</span>)                         <span class="Comment"># id is a unique identifier of an object</span></span>
<span class="line"><span class="anchor" id="line-5-24"></span><span class="Number">148293216</span></span>
<span class="line"><span class="anchor" id="line-6-24"></span>>>> <span class="ID">f</span>(<span class="ID">a</span>)</span>
<span class="line"><span class="anchor" id="line-7-23"></span><span class="Number">148293216</span></span>
</pre></div></div><span class="anchor" id="line-605"></span><p class="line867">
</p><h4 id="View_or_Shallow_Copy">View or Shallow Copy</h4>
<span class="anchor" id="line-606"></span><p class="line862">Different array objects can share the same data. The <tt class="backtick">view</tt> method creates a new array object that looks at the same data. <span class="anchor" id="line-607"></span><span class="anchor" id="line-608"></span></p><p class="line867"><span class="anchor" id="line-609"></span><span class="anchor" id="line-610"></span><span class="anchor" id="line-611"></span><span class="anchor" id="line-612"></span><span class="anchor" id="line-613"></span><span class="anchor" id="line-614"></span><span class="anchor" id="line-615"></span><span class="anchor" id="line-616"></span><span class="anchor" id="line-617"></span><span class="anchor" id="line-618"></span><span class="anchor" id="line-619"></span><span class="anchor" id="line-620"></span><span class="anchor" id="line-621"></span><span class="anchor" id="line-622"></span><span class="anchor" id="line-623"></span><span class="anchor" id="line-624"></span><span class="anchor" id="line-625"></span><span class="anchor" id="line-626"></span><span class="anchor" id="line-1-69"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-70c9cb0534c8246c9d52acdfca7374d2ce4b1366" lang="en"><span class="line"><span class="anchor" id="line-1-70"></span>>>> <span class="ID">c</span> = <span class="ID">a</span>.<span class="ID">view</span>()</span>
<span class="line"><span class="anchor" id="line-2-34"></span>>>> <span class="ID">c</span> <span class="ResWord">is</span> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-3-32"></span><span class="ResWord">False</span></span>
<span class="line"><span class="anchor" id="line-4-31"></span>>>> <span class="ID">c</span>.<span class="ID">base</span> <span class="ResWord">is</span> <span class="ID">a</span>                      <span class="Comment"># c is a view of the data owned by a</span></span>
<span class="line"><span class="anchor" id="line-5-25"></span><span class="ResWord">True</span></span>
<span class="line"><span class="anchor" id="line-6-25"></span>>>> <span class="ID">c</span>.<span class="ID">flags</span>.<span class="ID">owndata</span></span>
<span class="line"><span class="anchor" id="line-7-24"></span><span class="ResWord">False</span></span>
<span class="line"><span class="anchor" id="line-8-21"></span>>>></span>
<span class="line"><span class="anchor" id="line-9-20"></span>>>> <span class="ID">c</span>.<span class="ID">shape</span> = <span class="Number">2</span>,<span class="Number">6</span>                   <span class="Comment"># a's shape doesn't change</span></span>
<span class="line"><span class="anchor" id="line-10-19"></span>>>> <span class="ID">a</span>.<span class="ID">shape</span></span>
<span class="line"><span class="anchor" id="line-11-16"></span>(<span class="Number">3</span>, <span class="Number">4</span>)</span>
<span class="line"><span class="anchor" id="line-12-14"></span>>>> <span class="ID">c</span>[<span class="Number">0</span>,<span class="Number">4</span>] = <span class="Number">1234</span>                   <span class="Comment"># a's data changes</span></span>
<span class="line"><span class="anchor" id="line-13-14"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-14-12"></span><span class="ID">array</span>([[ <span class="Number">0</span>, <span class="Number">1</span>, <span class="Number">2</span>, <span class="Number">3</span>],</span>
<span class="line"><span class="anchor" id="line-15-9"></span>    [<span class="Number">1234</span>, <span class="Number">5</span>, <span class="Number">6</span>, <span class="Number">7</span>],</span>
<span class="line"><span class="anchor" id="line-16-8"></span>    [ <span class="Number">8</span>, <span class="Number">9</span>, <span class="Number">10</span>, <span class="Number">11</span>]])</span>
</pre></div></div><span class="anchor" id="line-627"></span><p class="line874">Slicing an array returns a view of it: <span class="anchor" id="line-628"></span><span class="anchor" id="line-629"></span></p><p class="line867"><span class="anchor" id="line-630"></span><span class="anchor" id="line-631"></span><span class="anchor" id="line-632"></span><span class="anchor" id="line-633"></span><span class="anchor" id="line-634"></span><span class="anchor" id="line-635"></span><span class="anchor" id="line-636"></span><span class="anchor" id="line-637"></span><span class="anchor" id="line-1-71"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-cbe5a4d6f027df93e6d55bacf4496f902850853c" lang="en"><span class="line"><span class="anchor" id="line-1-72"></span>>>> <span class="ID">s</span> = <span class="ID">a</span>[ : , <span class="Number">1</span>:<span class="Number">3</span>] <span class="Comment"># spaces added for clarity; could also be written "s = a[:,1:3]"</span></span>
<span class="line"><span class="anchor" id="line-2-35"></span>>>> <span class="ID">s</span>[:] = <span class="Number">10</span>       <span class="Comment"># s[:] is a view of s. Note the difference between s=10 and s[:]=10</span></span>
<span class="line"><span class="anchor" id="line-3-33"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-4-32"></span><span class="ID">array</span>([[ <span class="Number">0</span>, <span class="Number">10</span>, <span class="Number">10</span>, <span class="Number">3</span>],</span>
<span class="line"><span class="anchor" id="line-5-26"></span>    [<span class="Number">1234</span>, <span class="Number">10</span>, <span class="Number">10</span>, <span class="Number">7</span>],</span>
<span class="line"><span class="anchor" id="line-6-26"></span>    [ <span class="Number">8</span>, <span class="Number">10</span>, <span class="Number">10</span>, <span class="Number">11</span>]])</span>
</pre></div></div><span class="anchor" id="line-638"></span><p class="line867">
</p><h4 id="Deep_Copy">Deep Copy</h4>
<span class="anchor" id="line-639"></span><p class="line862">The <tt class="backtick">copy</tt> method makes a complete copy of the array and its data. <span class="anchor" id="line-640"></span><span class="anchor" id="line-641"></span></p><p class="line867"><span class="anchor" id="line-642"></span><span class="anchor" id="line-643"></span><span class="anchor" id="line-644"></span><span class="anchor" id="line-645"></span><span class="anchor" id="line-646"></span><span class="anchor" id="line-647"></span><span class="anchor" id="line-648"></span><span class="anchor" id="line-649"></span><span class="anchor" id="line-650"></span><span class="anchor" id="line-651"></span><span class="anchor" id="line-652"></span><span class="anchor" id="line-653"></span><span class="anchor" id="line-1-73"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-53089006e94d3bbc38bdfc003b736e9c2031500c" lang="en"><span class="line"><span class="anchor" id="line-1-74"></span>>>> <span class="ID">d</span> = <span class="ID">a</span>.<span class="ID">copy</span>()                      <span class="Comment"># a new array object with new data is created</span></span>
<span class="line"><span class="anchor" id="line-2-36"></span>>>> <span class="ID">d</span> <span class="ResWord">is</span> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-3-34"></span><span class="ResWord">False</span></span>
<span class="line"><span class="anchor" id="line-4-33"></span>>>> <span class="ID">d</span>.<span class="ID">base</span> <span class="ResWord">is</span> <span class="ID">a</span>                         <span class="Comment"># d doesn't share anything with a</span></span>
<span class="line"><span class="anchor" id="line-5-27"></span><span class="ResWord">False</span></span>
<span class="line"><span class="anchor" id="line-6-27"></span>>>> <span class="ID">d</span>[<span class="Number">0</span>,<span class="Number">0</span>] = <span class="Number">9999</span></span>
<span class="line"><span class="anchor" id="line-7-25"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-8-22"></span><span class="ID">array</span>([[ <span class="Number">0</span>, <span class="Number">10</span>, <span class="Number">10</span>, <span class="Number">3</span>],</span>
<span class="line"><span class="anchor" id="line-9-21"></span>    [<span class="Number">1234</span>, <span class="Number">10</span>, <span class="Number">10</span>, <span class="Number">7</span>],</span>
<span class="line"><span class="anchor" id="line-10-20"></span>    [ <span class="Number">8</span>, <span class="Number">10</span>, <span class="Number">10</span>, <span class="Number">11</span>]])</span>
</pre></div></div><span class="anchor" id="line-654"></span><p class="line867">
</p><h4 id="Functions_and_Methods_Overview">Functions and Methods Overview</h4>
<span class="anchor" id="line-655"></span><p class="line862">Here is a list of NumPy functions and methods names ordered in some categories. The names link to the <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html">Numpy_Example_List</a> so that you can see the functions in action. <span class="anchor" id="line-656"></span><span class="anchor" id="line-657"></span></p><dl><dt>Array Creation</dt><dd><p class="line891"><a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#arange">arange</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#array">array</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#copy">copy</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#empty">empty</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#empty_like">empty_like</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#eye">eye</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#fromfile">fromfile</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#fromfunction">fromfunction</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#identity">identity</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#linspace">linspace</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#logspace">logspace</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#mgrid">mgrid</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#ogrid">ogrid</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#ones">ones</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#ones_like">ones_like</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#r">r</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#zeros">zeros</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#zeros_like">zeros_like</a> <span class="anchor" id="line-658"></span></p></dd><dt>Conversions</dt><dd><p class="line891"><a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#astype">astype</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#atleast_1d">atleast 1d</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#atleast_2d">atleast 2d</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#atleast_3d">atleast 3d</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#mat">mat</a> <span class="anchor" id="line-659"></span></p></dd><dt>Manipulations</dt><dd><p class="line891"><a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#array_split">array split</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#column_stack">column stack</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#concatenate">concatenate</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#diagonal">diagonal</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#dsplit">dsplit</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#dstack">dstack</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#hsplit">hsplit</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#hstack">hstack</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#item">item</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#newaxis">newaxis</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#ravel">ravel</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#repeat">repeat</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#reshape">reshape</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#resize">resize</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#squeeze">squeeze</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#swapaxes">swapaxes</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#take">take</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#transpose">transpose</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#vsplit">vsplit</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#vstack">vstack</a> <span class="anchor" id="line-660"></span></p></dd><dt>Questions</dt><dd><p class="line891"><a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#all">all</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#any">any</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#nonzero">nonzero</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#where">where</a> <span class="anchor" id="line-661"></span></p></dd><dt>Ordering</dt><dd><p class="line891"><a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#argmax">argmax</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#argmin">argmin</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#argsort">argsort</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#max">max</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#min">min</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#ptp">ptp</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#searchsorted">searchsorted</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#sort">sort</a> <span class="anchor" id="line-662"></span></p></dd><dt>Operations</dt><dd><p class="line891"><a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#choose">choose</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#compress">compress</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#cumprod">cumprod</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#cumsum">cumsum</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#inner">inner</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#fill">fill</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#imag">imag</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#prod">prod</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#put">put</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#putmask">putmask</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#real">real</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#sum">sum</a> <span class="anchor" id="line-663"></span></p></dd><dt>Basic Statistics</dt><dd><p class="line891"><a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#cov">cov</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#mean">mean</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#std">std</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#var">var</a> <span class="anchor" id="line-664"></span></p></dd><dt>Basic Linear Algebra</dt><dd><p class="line891"><a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#cross">cross</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#dot">dot</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#outer">outer</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#svd">svd</a>, <a href="https://scipy.github.io/old-wiki/pages/Numpy_Example_List.html#vdot">vdot</a> <span class="anchor" id="line-665"></span></p></dd></dl><p class="line867">
</p><h2 id="Less_Basic">Less Basic</h2>
<span class="anchor" id="line-666"></span><span class="anchor" id="line-667"></span><span class="anchor" id="line-668"></span><p class="line867">
</p><h4 id="Broadcasting_rules">Broadcasting rules</h4>
<span class="anchor" id="line-669"></span><p class="line874">Broadcasting allows universal functions to deal in a meaningful way with inputs that do not have exactly the same shape. <span class="anchor" id="line-670"></span><span class="anchor" id="line-671"></span></p><p class="line874">The first rule of broadcasting is that if all input arrays do not have the same number of dimensions, a "1" will be repeatedly prepended to the shapes of the smaller arrays until all the arrays have the same number of dimensions. <span class="anchor" id="line-672"></span><span class="anchor" id="line-673"></span></p><p class="line874">The second rule of broadcasting ensures that arrays with a size of 1 along a particular dimension act as if they had the size of the array with the largest shape along that dimension. The value of the array element is assumed to be the same along that dimension for the "broadcast" array. <span class="anchor" id="line-674"></span><span class="anchor" id="line-675"></span></p><p class="line862">After application of the broadcasting rules, the sizes of all arrays must match. More details can be found in <a href="https://scipy.github.io/old-wiki/pages/EricsBroadcastingDoc.html">this documentation</a>. <span class="anchor" id="line-676"></span><span class="anchor" id="line-677"></span></p><p class="line867">
</p><h2 id="Fancy_indexing_and_index_tricks">Fancy indexing and index tricks</h2>
<span class="anchor" id="line-678"></span><p class="line867">NumPy offers more indexing facilities than regular Python sequences. In addition to indexing by integers and slices, as we saw before, arrays can be indexed by arrays of integers and arrays of booleans. <span class="anchor" id="line-679"></span><span class="anchor" id="line-680"></span></p><p class="line867">
</p><h4 id="Indexing_with_Arrays_of_Indices">Indexing with Arrays of Indices</h4>
<span class="anchor" id="line-681"></span><p class="line867"><span class="anchor" id="line-682"></span><span class="anchor" id="line-683"></span><span class="anchor" id="line-684"></span><span class="anchor" id="line-685"></span><span class="anchor" id="line-686"></span><span class="anchor" id="line-687"></span><span class="anchor" id="line-688"></span><span class="anchor" id="line-689"></span><span class="anchor" id="line-690"></span><span class="anchor" id="line-691"></span><span class="anchor" id="line-692"></span><span class="anchor" id="line-1-75"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-bf3fd737324d219e925ce168187330ae37bab2dd" lang="en"><span class="line"><span class="anchor" id="line-1-76"></span>>>> <span class="ID">a</span> = <span class="ID">arange</span>(<span class="Number">12</span>)**<span class="Number">2</span>                      <span class="Comment"># the first 12 square numbers</span></span>
<span class="line"><span class="anchor" id="line-2-37"></span>>>> <span class="ID">i</span> = <span class="ID">array</span>( [ <span class="Number">1</span>,<span class="Number">1</span>,<span class="Number">3</span>,<span class="Number">8</span>,<span class="Number">5</span> ] )             <span class="Comment"># an array of indices</span></span>
<span class="line"><span class="anchor" id="line-3-35"></span>>>> <span class="ID">a</span>[<span class="ID">i</span>]                                     <span class="Comment"># the elements of a at the positions i</span></span>
<span class="line"><span class="anchor" id="line-4-34"></span><span class="ID">array</span>([ <span class="Number">1</span>,<span class="Number">1</span>,<span class="Number">9</span>, <span class="Number">64</span>, <span class="Number">25</span>])</span>
<span class="line"><span class="anchor" id="line-5-28"></span>>>></span>
<span class="line"><span class="anchor" id="line-6-28"></span>>>> <span class="ID">j</span> = <span class="ID">array</span>( [ [ <span class="Number">3</span>, <span class="Number">4</span>], [ <span class="Number">9</span>, <span class="Number">7</span> ] ] )       <span class="Comment"># a bidimensional array of indices</span></span>
<span class="line"><span class="anchor" id="line-7-26"></span>>>> <span class="ID">a</span>[<span class="ID">j</span>]                                     <span class="Comment"># the same shape as j</span></span>
<span class="line"><span class="anchor" id="line-8-23"></span><span class="ID">array</span>([[ <span class="Number">9</span>, <span class="Number">16</span>],</span>
<span class="line"><span class="anchor" id="line-9-22"></span>    [<span class="Number">81</span>, <span class="Number">49</span>]])</span>
</pre></div></div><span class="anchor" id="line-693"></span><p class="line862">When the indexed array <tt class="backtick">a</tt> is multidimensional, a single array of indices refers to the first dimension of <tt class="backtick">a</tt>. The following example shows this behavior by converting an image of labels into a color image using a palette. <span class="anchor" id="line-694"></span><span class="anchor" id="line-695"></span></p><p class="line867"><span class="anchor" id="line-696"></span><span class="anchor" id="line-697"></span><span class="anchor" id="line-698"></span><span class="anchor" id="line-699"></span><span class="anchor" id="line-700"></span><span class="anchor" id="line-701"></span><span class="anchor" id="line-702"></span><span class="anchor" id="line-703"></span><span class="anchor" id="line-704"></span><span class="anchor" id="line-705"></span><span class="anchor" id="line-706"></span><span class="anchor" id="line-707"></span><span class="anchor" id="line-708"></span><span class="anchor" id="line-709"></span><span class="anchor" id="line-710"></span><span class="anchor" id="line-711"></span><span class="anchor" id="line-712"></span><span class="anchor" id="line-713"></span><span class="anchor" id="line-1-77"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-8d7825a313fe52ddd2c5a9b6c338afea41ec739d" lang="en"><span class="line"><span class="anchor" id="line-1-78"></span>>>> <span class="ID">palette</span> = <span class="ID">array</span>( [ [<span class="Number">0</span>,<span class="Number">0</span>,<span class="Number">0</span>],             <span class="Comment"># black</span></span>
<span class="line"><span class="anchor" id="line-2-38"></span>...                [<span class="Number">255</span>,<span class="Number">0</span>,<span class="Number">0</span>],          <span class="Comment"># red</span></span>
<span class="line"><span class="anchor" id="line-3-36"></span>...                [<span class="Number">0</span>,<span class="Number">255</span>,<span class="Number">0</span>],          <span class="Comment"># green</span></span>
<span class="line"><span class="anchor" id="line-4-35"></span>...                [<span class="Number">0</span>,<span class="Number">0</span>,<span class="Number">255</span>],          <span class="Comment"># blue</span></span>
<span class="line"><span class="anchor" id="line-5-29"></span>...                [<span class="Number">255</span>,<span class="Number">255</span>,<span class="Number">255</span>] ] )    <span class="Comment"># white</span></span>
<span class="line"><span class="anchor" id="line-6-29"></span>>>> <span class="ID">image</span> = <span class="ID">array</span>( [ [ <span class="Number">0</span>, <span class="Number">1</span>, <span class="Number">2</span>, <span class="Number">0</span> ],       <span class="Comment"># each value corresponds to a color in the palette</span></span>
<span class="line"><span class="anchor" id="line-7-27"></span>...                [ <span class="Number">0</span>, <span class="Number">3</span>, <span class="Number">4</span>, <span class="Number">0</span> ]] )</span>
<span class="line"><span class="anchor" id="line-8-24"></span>>>> <span class="ID">palette</span>[<span class="ID">image</span>]                         <span class="Comment"># the (2,4,3) color image</span></span>
<span class="line"><span class="anchor" id="line-9-23"></span><span class="ID">array</span>([[[<span class="Number">0</span>, <span class="Number">0</span>, <span class="Number">0</span>],</span>
<span class="line"><span class="anchor" id="line-10-21"></span>    [<span class="Number">255</span>, <span class="Number">0</span>, <span class="Number">0</span>],</span>
<span class="line"><span class="anchor" id="line-11-17"></span>    [<span class="Number">0</span>, <span class="Number">255</span>, <span class="Number">0</span>],</span>
<span class="line"><span class="anchor" id="line-12-15"></span>    [<span class="Number">0</span>, <span class="Number">0</span>, <span class="Number">0</span>]],</span>
<span class="line"><span class="anchor" id="line-13-15"></span>    [[<span class="Number">0</span>, <span class="Number">0</span>, <span class="Number">0</span>],</span>
<span class="line"><span class="anchor" id="line-14-13"></span>    [<span class="Number">0</span>, <span class="Number">0</span>, <span class="Number">255</span>],</span>
<span class="line"><span class="anchor" id="line-15-10"></span>    [<span class="Number">255</span>, <span class="Number">255</span>, <span class="Number">255</span>],</span>
<span class="line"><span class="anchor" id="line-16-9"></span>    [<span class="Number">0</span>, <span class="Number">0</span>, <span class="Number">0</span>]]])</span>
</pre></div></div><span class="anchor" id="line-714"></span><p class="line874">We can also give indexes for more than one dimension. The arrays of indices for each dimension must have the same shape. <span class="anchor" id="line-715"></span><span class="anchor" id="line-716"></span></p><p class="line867"><span class="anchor" id="line-717"></span><span class="anchor" id="line-718"></span><span class="anchor" id="line-719"></span><span class="anchor" id="line-720"></span><span class="anchor" id="line-721"></span><span class="anchor" id="line-722"></span><span class="anchor" id="line-723"></span><span class="anchor" id="line-724"></span><span class="anchor" id="line-725"></span><span class="anchor" id="line-726"></span><span class="anchor" id="line-727"></span><span class="anchor" id="line-728"></span><span class="anchor" id="line-729"></span><span class="anchor" id="line-730"></span><span class="anchor" id="line-731"></span><span class="anchor" id="line-732"></span><span class="anchor" id="line-733"></span><span class="anchor" id="line-734"></span><span class="anchor" id="line-735"></span><span class="anchor" id="line-736"></span><span class="anchor" id="line-737"></span><span class="anchor" id="line-738"></span><span class="anchor" id="line-739"></span><span class="anchor" id="line-740"></span><span class="anchor" id="line-741"></span><span class="anchor" id="line-742"></span><span class="anchor" id="line-743"></span><span class="anchor" id="line-1-79"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-09d9c7082348bd8dc8d74192502eca501c9e9b52" lang="en"><span class="line"><span class="anchor" id="line-1-80"></span>>>> <span class="ID">a</span> = <span class="ID">arange</span>(<span class="Number">12</span>).<span class="ID">reshape</span>(<span class="Number">3</span>,<span class="Number">4</span>)</span>
<span class="line"><span class="anchor" id="line-2-39"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-3-37"></span><span class="ID">array</span>([[ <span class="Number">0</span>,<span class="Number">1</span>,<span class="Number">2</span>,<span class="Number">3</span>],</span>
<span class="line"><span class="anchor" id="line-4-36"></span>    [ <span class="Number">4</span>,<span class="Number">5</span>,<span class="Number">6</span>,<span class="Number">7</span>],</span>
<span class="line"><span class="anchor" id="line-5-30"></span>    [ <span class="Number">8</span>,<span class="Number">9</span>, <span class="Number">10</span>, <span class="Number">11</span>]])</span>
<span class="line"><span class="anchor" id="line-6-30"></span>>>> <span class="ID">i</span> = <span class="ID">array</span>( [ [<span class="Number">0</span>,<span class="Number">1</span>],                      <span class="Comment"># indices for the first dim of a</span></span>
<span class="line"><span class="anchor" id="line-7-28"></span>...          [<span class="Number">1</span>,<span class="Number">2</span>] ] )</span>
<span class="line"><span class="anchor" id="line-8-25"></span>>>> <span class="ID">j</span> = <span class="ID">array</span>( [ [<span class="Number">2</span>,<span class="Number">1</span>],                      <span class="Comment"># indices for the second dim</span></span>
<span class="line"><span class="anchor" id="line-9-24"></span>...          [<span class="Number">3</span>,<span class="Number">3</span>] ] )</span>
<span class="line"><span class="anchor" id="line-10-22"></span>>>></span>
<span class="line"><span class="anchor" id="line-11-18"></span>>>> <span class="ID">a</span>[<span class="ID">i</span>,<span class="ID">j</span>]                                  <span class="Comment"># i and j must have equal shape</span></span>
<span class="line"><span class="anchor" id="line-12-16"></span><span class="ID">array</span>([[ <span class="Number">2</span>,<span class="Number">5</span>],</span>
<span class="line"><span class="anchor" id="line-13-16"></span>    [ <span class="Number">7</span>, <span class="Number">11</span>]])</span>
<span class="line"><span class="anchor" id="line-14-14"></span>>>></span>
<span class="line"><span class="anchor" id="line-15-11"></span>>>> <span class="ID">a</span>[<span class="ID">i</span>,<span class="Number">2</span>]</span>
<span class="line"><span class="anchor" id="line-16-10"></span><span class="ID">array</span>([[ <span class="Number">2</span>,<span class="Number">6</span>],</span>
<span class="line"><span class="anchor" id="line-17-4"></span>    [ <span class="Number">6</span>, <span class="Number">10</span>]])</span>
<span class="line"><span class="anchor" id="line-18-4"></span>>>></span>
<span class="line"><span class="anchor" id="line-19-4"></span>>>> <span class="ID">a</span>[:,<span class="ID">j</span>]                                  <span class="Comment"># i.e., a[ : , j]</span></span>
<span class="line"><span class="anchor" id="line-20-3"></span><span class="ID">array</span>([[[ <span class="Number">2</span>,<span class="Number">1</span>],</span>
<span class="line"><span class="anchor" id="line-21-2"></span>    [ <span class="Number">3</span>,<span class="Number">3</span>]],</span>
<span class="line"><span class="anchor" id="line-22-2"></span>    [[ <span class="Number">6</span>,<span class="Number">5</span>],</span>
<span class="line"><span class="anchor" id="line-23-2"></span>    [ <span class="Number">7</span>,<span class="Number">7</span>]],</span>
<span class="line"><span class="anchor" id="line-24-1"></span>    [[<span class="Number">10</span>,<span class="Number">9</span>],</span>
<span class="line"><span class="anchor" id="line-25-1"></span>    [<span class="Number">11</span>, <span class="Number">11</span>]]])</span>
</pre></div></div><span class="anchor" id="line-744"></span><p class="line862">Naturally, we can put <tt class="backtick">i</tt> and <tt class="backtick">j</tt> in a sequence (say a list) and then do the indexing with the list. <span class="anchor" id="line-745"></span><span class="anchor" id="line-746"></span></p><p class="line867"><span class="anchor" id="line-747"></span><span class="anchor" id="line-748"></span><span class="anchor" id="line-749"></span><span class="anchor" id="line-750"></span><span class="anchor" id="line-751"></span><span class="anchor" id="line-752"></span><span class="anchor" id="line-1-81"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-bfb4985b58c2e3993d826b1a6aa96ef6384631e2" lang="en"><span class="line"><span class="anchor" id="line-1-82"></span>>>> <span class="ID">l</span> = [<span class="ID">i</span>,<span class="ID">j</span>]</span>
<span class="line"><span class="anchor" id="line-2-40"></span>>>> <span class="ID">a</span>[<span class="ID">l</span>]                                     <span class="Comment"># equivalent to a</span></span>
<span class="line"><span class="anchor" id="line-3-38"></span><span class="ID">array</span>([[ <span class="Number">2</span>,<span class="Number">5</span>],</span>
<span class="line"><span class="anchor" id="line-4-37"></span>    [ <span class="Number">7</span>, <span class="Number">11</span>]])</span>
</pre></div></div><span class="anchor" id="line-753"></span><p class="line862">However, we can not do this by putting <tt class="backtick">i</tt> and <tt class="backtick">j</tt> into an array, because this array will be interpreted as indexing the first dimension of a. <span class="anchor" id="line-754"></span><span class="anchor" id="line-755"></span></p><p class="line867"><span class="anchor" id="line-756"></span><span class="anchor" id="line-757"></span><span class="anchor" id="line-758"></span><span class="anchor" id="line-759"></span><span class="anchor" id="line-760"></span><span class="anchor" id="line-761"></span><span class="anchor" id="line-762"></span><span class="anchor" id="line-763"></span><span class="anchor" id="line-764"></span><span class="anchor" id="line-765"></span><span class="anchor" id="line-766"></span><span class="anchor" id="line-1-83"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-c39d63f33c74cc7dd390d7795d5da6d5f20ab707" lang="en"><span class="line"><span class="anchor" id="line-1-84"></span>>>> <span class="ID">s</span> = <span class="ID">array</span>( [<span class="ID">i</span>,<span class="ID">j</span>] )</span>
<span class="line"><span class="anchor" id="line-2-41"></span>>>> <span class="ID">a</span>[<span class="ID">s</span>]                                     <span class="Comment"># not what we want</span></span>
<span class="line"><span class="anchor" id="line-3-39"></span><span class="ID">Traceback</span> (<span class="ID">most</span> <span class="ID">recent</span> <span class="ID">call</span> <span class="ID">last</span>):</span>
<span class="line"><span class="anchor" id="line-4-38"></span><span class="ID">File</span> <span class="String">"</span><span class="String"><stdin></span><span class="String">"</span>, <span class="ID">line</span> <span class="Number">1</span>, <span class="ResWord">in</span> ?</span>
<span class="line"><span class="anchor" id="line-5-31"></span><span class="ID">IndexError</span>: <span class="ID">index</span> (<span class="Number">3</span>) <span class="ID">out</span> <span class="ID">of</span> <span class="ResWord">range</span> (<span class="Number">0</span><=<span class="ID">index</span><=<span class="Number">2</span>) <span class="ResWord">in</span> <span class="ID">dimension</span> <span class="Number">0</span></span>
<span class="line"><span class="anchor" id="line-6-31"></span>>>></span>
<span class="line"><span class="anchor" id="line-7-29"></span>>>> <span class="ID">a</span>[<span class="ResWord">tuple</span>(<span class="ID">s</span>)]                            <span class="Comment"># same as a</span></span>
<span class="line"><span class="anchor" id="line-8-26"></span><span class="ID">array</span>([[ <span class="Number">2</span>,<span class="Number">5</span>],</span>
<span class="line"><span class="anchor" id="line-9-25"></span>    [ <span class="Number">7</span>, <span class="Number">11</span>]])</span>
</pre></div></div><span class="anchor" id="line-767"></span><p class="line874">Another common use of indexing with arrays is the search of the maximum value of time-dependent series : <span class="anchor" id="line-768"></span><span class="anchor" id="line-769"></span></p><p class="line867"><span class="anchor" id="line-770"></span><span class="anchor" id="line-771"></span><span class="anchor" id="line-772"></span><span class="anchor" id="line-773"></span><span class="anchor" id="line-774"></span><span class="anchor" id="line-775"></span><span class="anchor" id="line-776"></span><span class="anchor" id="line-777"></span><span class="anchor" id="line-778"></span><span class="anchor" id="line-779"></span><span class="anchor" id="line-780"></span><span class="anchor" id="line-781"></span><span class="anchor" id="line-782"></span><span class="anchor" id="line-783"></span><span class="anchor" id="line-784"></span><span class="anchor" id="line-785"></span><span class="anchor" id="line-786"></span><span class="anchor" id="line-787"></span><span class="anchor" id="line-788"></span><span class="anchor" id="line-789"></span><span class="anchor" id="line-790"></span><span class="anchor" id="line-791"></span><span class="anchor" id="line-792"></span><span class="anchor" id="line-793"></span><span class="anchor" id="line-794"></span><span class="anchor" id="line-795"></span><span class="anchor" id="line-796"></span><span class="anchor" id="line-797"></span><span class="anchor" id="line-1-85"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-32794fa93cb64d6e36f18885abae085ea67c525b" lang="en"><span class="line"><span class="anchor" id="line-1-86"></span>>>> <span class="ID">time</span> = <span class="ID">linspace</span>(<span class="Number">20</span>, <span class="Number">145</span>, <span class="Number">5</span>)             <span class="Comment"># time scale</span></span>
<span class="line"><span class="anchor" id="line-2-42"></span>>>> <span class="ID">data</span> = <span class="ID">sin</span>(<span class="ID">arange</span>(<span class="Number">20</span>)).<span class="ID">reshape</span>(<span class="Number">5</span>,<span class="Number">4</span>)       <span class="Comment"># 4 time-dependent series</span></span>
<span class="line"><span class="anchor" id="line-3-40"></span>>>> <span class="ID">time</span></span>
<span class="line"><span class="anchor" id="line-4-39"></span><span class="ID">array</span>([<span class="Number">20.</span>, <span class="Number">51.25</span>, <span class="Number">82.5</span> ,<span class="Number">113.75</span>,<span class="Number">145.</span>])</span>
<span class="line"><span class="anchor" id="line-5-32"></span>>>> <span class="ID">data</span></span>
<span class="line"><span class="anchor" id="line-6-32"></span><span class="ID">array</span>([[ <span class="Number">0.</span>    ,<span class="Number">0.84147098</span>,<span class="Number">0.90929743</span>,<span class="Number">0.14112001</span>],</span>
<span class="line"><span class="anchor" id="line-7-30"></span>    [-<span class="Number">0.7568025</span> , -<span class="Number">0.95892427</span>, -<span class="Number">0.2794155</span> ,<span class="Number">0.6569866</span> ],</span>
<span class="line"><span class="anchor" id="line-8-27"></span>    [ <span class="Number">0.98935825</span>,<span class="Number">0.41211849</span>, -<span class="Number">0.54402111</span>, -<span class="Number">0.99999021</span>],</span>
<span class="line"><span class="anchor" id="line-9-26"></span>    [-<span class="Number">0.53657292</span>,<span class="Number">0.42016704</span>,<span class="Number">0.99060736</span>,<span class="Number">0.65028784</span>],</span>
<span class="line"><span class="anchor" id="line-10-23"></span>    [-<span class="Number">0.28790332</span>, -<span class="Number">0.96139749</span>, -<span class="Number">0.75098725</span>,<span class="Number">0.14987721</span>]])</span>
<span class="line"><span class="anchor" id="line-11-19"></span>>>></span>
<span class="line"><span class="anchor" id="line-12-17"></span>>>> <span class="ID">ind</span> = <span class="ID">data</span>.<span class="ID">argmax</span>(<span class="ID">axis</span>=<span class="Number">0</span>)                <span class="Comment"># index of the maxima for each series</span></span>
<span class="line"><span class="anchor" id="line-13-17"></span>>>> <span class="ID">ind</span></span>
<span class="line"><span class="anchor" id="line-14-15"></span><span class="ID">array</span>([<span class="Number">2</span>, <span class="Number">0</span>, <span class="Number">3</span>, <span class="Number">1</span>])</span>
<span class="line"><span class="anchor" id="line-15-12"></span>>>></span>
<span class="line"><span class="anchor" id="line-16-11"></span>>>> <span class="ID">time_max</span> = <span class="ID">time</span>[ <span class="ID">ind</span>]                   <span class="Comment"># times corresponding to the maxima</span></span>
<span class="line"><span class="anchor" id="line-17-5"></span>>>></span>
<span class="line"><span class="anchor" id="line-18-5"></span>>>> <span class="ID">data_max</span> = <span class="ID">data</span>[<span class="ID">ind</span>, <span class="ResWord">xrange</span>(<span class="ID">data</span>.<span class="ID">shape</span>[<span class="Number">1</span>])] <span class="Comment"># => data,0], data,1]...</span></span>
<span class="line"><span class="anchor" id="line-19-5"></span>>>></span>
<span class="line"><span class="anchor" id="line-20-4"></span>>>> <span class="ID">time_max</span></span>
<span class="line"><span class="anchor" id="line-21-3"></span><span class="ID">array</span>([<span class="Number">82.5</span> , <span class="Number">20.</span>,<span class="Number">113.75</span>, <span class="Number">51.25</span>])</span>
<span class="line"><span class="anchor" id="line-22-3"></span>>>> <span class="ID">data_max</span></span>
<span class="line"><span class="anchor" id="line-23-3"></span><span class="ID">array</span>([ <span class="Number">0.98935825</span>,<span class="Number">0.84147098</span>,<span class="Number">0.99060736</span>,<span class="Number">0.6569866</span> ])</span>
<span class="line"><span class="anchor" id="line-24-2"></span>>>></span>
<span class="line"><span class="anchor" id="line-25-2"></span>>>> <span class="ResWord">all</span>(<span class="ID">data_max</span> == <span class="ID">data</span>.<span class="ID">max</span>(<span class="ID">axis</span>=<span class="Number">0</span>))</span>
<span class="line"><span class="anchor" id="line-26-1"></span><span class="ResWord">True</span></span>
</pre></div></div><span class="anchor" id="line-798"></span><p class="line874">You can also use indexing with arrays as a target to assign to: <span class="anchor" id="line-799"></span><span class="anchor" id="line-800"></span></p><p class="line867"><span class="anchor" id="line-801"></span><span class="anchor" id="line-802"></span><span class="anchor" id="line-803"></span><span class="anchor" id="line-804"></span><span class="anchor" id="line-805"></span><span class="anchor" id="line-806"></span><span class="anchor" id="line-807"></span><span class="anchor" id="line-808"></span><span class="anchor" id="line-1-87"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-746f8c12d7ab1ac877ffbf6a5327729300dd9cff" lang="en"><span class="line"><span class="anchor" id="line-1-88"></span>>>> <span class="ID">a</span> = <span class="ID">arange</span>(<span class="Number">5</span>)</span>
<span class="line"><span class="anchor" id="line-2-43"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-3-41"></span><span class="ID">array</span>([<span class="Number">0</span>, <span class="Number">1</span>, <span class="Number">2</span>, <span class="Number">3</span>, <span class="Number">4</span>])</span>
<span class="line"><span class="anchor" id="line-4-40"></span>>>> <span class="ID">a</span>[[<span class="Number">1</span>,<span class="Number">3</span>,<span class="Number">4</span>]] = <span class="Number">0</span></span>
<span class="line"><span class="anchor" id="line-5-33"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-6-33"></span><span class="ID">array</span>([<span class="Number">0</span>, <span class="Number">0</span>, <span class="Number">2</span>, <span class="Number">0</span>, <span class="Number">0</span>])</span>
</pre></div></div><span class="anchor" id="line-809"></span><p class="line874">However, when the list of indices contains repetitions, the assignment is done several times, leaving behind the last value: <span class="anchor" id="line-810"></span><span class="anchor" id="line-811"></span></p><p class="line867"><span class="anchor" id="line-812"></span><span class="anchor" id="line-813"></span><span class="anchor" id="line-814"></span><span class="anchor" id="line-815"></span><span class="anchor" id="line-816"></span><span class="anchor" id="line-817"></span><span class="anchor" id="line-1-89"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-a68087e747cefc09da3089080c8b5a7ea6bacfa3" lang="en"><span class="line"><span class="anchor" id="line-1-90"></span>>>> <span class="ID">a</span> = <span class="ID">arange</span>(<span class="Number">5</span>)</span>
<span class="line"><span class="anchor" id="line-2-44"></span>>>> <span class="ID">a</span>[[<span class="Number">0</span>,<span class="Number">0</span>,<span class="Number">2</span>]]=[<span class="Number">1</span>,<span class="Number">2</span>,<span class="Number">3</span>]</span>
<span class="line"><span class="anchor" id="line-3-42"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-4-41"></span><span class="ID">array</span>([<span class="Number">2</span>, <span class="Number">1</span>, <span class="Number">3</span>, <span class="Number">3</span>, <span class="Number">4</span>])</span>
</pre></div></div><span class="anchor" id="line-818"></span><p class="line862">This is reasonable enough, but watch out if you want to use Python's <tt class="backtick">+=</tt> construct, as it may not do what you expect: <span class="anchor" id="line-819"></span><span class="anchor" id="line-820"></span></p><p class="line867"><span class="anchor" id="line-821"></span><span class="anchor" id="line-822"></span><span class="anchor" id="line-823"></span><span class="anchor" id="line-824"></span><span class="anchor" id="line-825"></span><span class="anchor" id="line-826"></span><span class="anchor" id="line-1-91"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-be70a5387a8d1f1412e51c75c9a920391a88458e" lang="en"><span class="line"><span class="anchor" id="line-1-92"></span>>>> <span class="ID">a</span> = <span class="ID">arange</span>(<span class="Number">5</span>)</span>
<span class="line"><span class="anchor" id="line-2-45"></span>>>> <span class="ID">a</span>[[<span class="Number">0</span>,<span class="Number">0</span>,<span class="Number">2</span>]]+=<span class="Number">1</span></span>
<span class="line"><span class="anchor" id="line-3-43"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-4-42"></span><span class="ID">array</span>([<span class="Number">1</span>, <span class="Number">1</span>, <span class="Number">3</span>, <span class="Number">3</span>, <span class="Number">4</span>])</span>
</pre></div></div><span class="anchor" id="line-827"></span><p class="line874">Even though 0 occurs twice in the list of indices, the 0th element is only incremented once. This is because Python requires "a+=1" to be equivalent to "a=a+1". <span class="anchor" id="line-828"></span><span class="anchor" id="line-829"></span></p><p class="line867">
</p><h4 id="Indexing_with_Boolean_Arrays">Indexing with Boolean Arrays</h4>
<span class="anchor" id="line-830"></span><p class="line874">When we index arrays with arrays of (integer) indices we are providing the list of indices to pick. With boolean indices the approach is different; we explicitly choose which items in the array we want and which ones we don't. <span class="anchor" id="line-831"></span><span class="anchor" id="line-832"></span></p><p class="line862">The most natural way one can think of for boolean indexing is to use boolean arrays that have <em>the same shape</em> as the original array: <span class="anchor" id="line-833"></span><span class="anchor" id="line-834"></span></p><p class="line867"><span class="anchor" id="line-835"></span><span class="anchor" id="line-836"></span><span class="anchor" id="line-837"></span><span class="anchor" id="line-838"></span><span class="anchor" id="line-839"></span><span class="anchor" id="line-840"></span><span class="anchor" id="line-841"></span><span class="anchor" id="line-842"></span><span class="anchor" id="line-843"></span><span class="anchor" id="line-844"></span><span class="anchor" id="line-1-93"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-3cfbc7de5712f5af64c42d7cfc6024edbd402f5a" lang="en"><span class="line"><span class="anchor" id="line-1-94"></span>>>> <span class="ID">a</span> = <span class="ID">arange</span>(<span class="Number">12</span>).<span class="ID">reshape</span>(<span class="Number">3</span>,<span class="Number">4</span>)</span>
<span class="line"><span class="anchor" id="line-2-46"></span>>>> <span class="ID">b</span> = <span class="ID">a</span> > <span class="Number">4</span></span>
<span class="line"><span class="anchor" id="line-3-44"></span>>>> <span class="ID">b</span>                                        <span class="Comment"># b is a boolean with a's shape</span></span>
<span class="line"><span class="anchor" id="line-4-43"></span><span class="ID">array</span>([[<span class="ResWord">False</span>, <span class="ResWord">False</span>, <span class="ResWord">False</span>, <span class="ResWord">False</span>],</span>
<span class="line"><span class="anchor" id="line-5-34"></span>    [<span class="ResWord">False</span>, <span class="ResWord">True</span>, <span class="ResWord">True</span>, <span class="ResWord">True</span>],</span>
<span class="line"><span class="anchor" id="line-6-34"></span>    [<span class="ResWord">True</span>, <span class="ResWord">True</span>, <span class="ResWord">True</span>, <span class="ResWord">True</span>]], <span class="ID">dtype</span>=<span class="ResWord">bool</span>)</span>
<span class="line"><span class="anchor" id="line-7-31"></span>>>> <span class="ID">a</span>[<span class="ID">b</span>]                                     <span class="Comment"># 1d array with the selected elements</span></span>
<span class="line"><span class="anchor" id="line-8-28"></span><span class="ID">array</span>([ <span class="Number">5</span>,<span class="Number">6</span>,<span class="Number">7</span>,<span class="Number">8</span>,<span class="Number">9</span>, <span class="Number">10</span>, <span class="Number">11</span>])</span>
</pre></div></div><span class="anchor" id="line-845"></span><p class="line874">This property can be very useful in assignments: <span class="anchor" id="line-846"></span><span class="anchor" id="line-847"></span></p><p class="line867"><span class="anchor" id="line-848"></span><span class="anchor" id="line-849"></span><span class="anchor" id="line-850"></span><span class="anchor" id="line-851"></span><span class="anchor" id="line-852"></span><span class="anchor" id="line-853"></span><span class="anchor" id="line-854"></span><span class="anchor" id="line-1-95"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-f481bdf09e2e249220f82919f7fd747d6e8571fc" lang="en"><span class="line"><span class="anchor" id="line-1-96"></span>>>> <span class="ID">a</span>[<span class="ID">b</span>] = <span class="Number">0</span>                               <span class="Comment"># All elements of 'a' higher than 4 become 0</span></span>
<span class="line"><span class="anchor" id="line-2-47"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-3-45"></span><span class="ID">array</span>([[<span class="Number">0</span>, <span class="Number">1</span>, <span class="Number">2</span>, <span class="Number">3</span>],</span>
<span class="line"><span class="anchor" id="line-4-44"></span>    [<span class="Number">4</span>, <span class="Number">0</span>, <span class="Number">0</span>, <span class="Number">0</span>],</span>
<span class="line"><span class="anchor" id="line-5-35"></span>    [<span class="Number">0</span>, <span class="Number">0</span>, <span class="Number">0</span>, <span class="Number">0</span>]])</span>
</pre></div></div><span class="anchor" id="line-855"></span><p class="line862">You can look at the <a href="https://scipy.github.io/old-wiki/pages/Tentative_NumPy_Tutorial/Mandelbrot_Set_Example.html">Mandelbrot set example</a> to see how to use boolean indexing to generate an image of the <ahref="http://en.wikipedia.org/wiki/Mandelbrot_set">Mandelbrot set</a>. <span class="anchor" id="line-856"></span><span class="anchor" id="line-857"></span></p><p class="line874">The second way of indexing with booleans is more similar to integer indexing; for each dimension of the array we give a 1D boolean array selecting the slices we want. <span class="anchor" id="line-858"></span><span class="anchor" id="line-859"></span></p><p class="line867"><span class="anchor" id="line-860"></span><span class="anchor" id="line-861"></span><span class="anchor" id="line-862"></span><span class="anchor" id="line-863"></span><span class="anchor" id="line-864"></span><span class="anchor" id="line-865"></span><span class="anchor" id="line-866"></span><span class="anchor" id="line-867"></span><span class="anchor" id="line-868"></span><span class="anchor" id="line-869"></span><span class="anchor" id="line-870"></span><span class="anchor" id="line-871"></span><span class="anchor" id="line-872"></span><span class="anchor" id="line-873"></span><span class="anchor" id="line-874"></span><span class="anchor" id="line-875"></span><span class="anchor" id="line-876"></span><span class="anchor" id="line-877"></span><span class="anchor" id="line-878"></span><span class="anchor" id="line-879"></span><span class="anchor" id="line-880"></span><span class="anchor" id="line-1-97"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-e2b7309c40a565c716c6e64af7bd202d8721ffe4" lang="en"><span class="line"><span class="anchor" id="line-1-98"></span>>>> <span class="ID">a</span> = <span class="ID">arange</span>(<span class="Number">12</span>).<span class="ID">reshape</span>(<span class="Number">3</span>,<span class="Number">4</span>)</span>
<span class="line"><span class="anchor" id="line-2-48"></span>>>> <span class="ID">b1</span> = <span class="ID">array</span>([<span class="ResWord">False</span>,<span class="ResWord">True</span>,<span class="ResWord">True</span>])          <span class="Comment"># first dim selection</span></span>
<span class="line"><span class="anchor" id="line-3-46"></span>>>> <span class="ID">b2</span> = <span class="ID">array</span>([<span class="ResWord">True</span>,<span class="ResWord">False</span>,<span class="ResWord">True</span>,<span class="ResWord">False</span>])    <span class="Comment"># second dim selection</span></span>
<span class="line"><span class="anchor" id="line-4-45"></span>>>></span>
<span class="line"><span class="anchor" id="line-5-36"></span>>>> <span class="ID">a</span>[<span class="ID">b1</span>,:]                               <span class="Comment"># selecting rows</span></span>
<span class="line"><span class="anchor" id="line-6-35"></span><span class="ID">array</span>([[ <span class="Number">4</span>,<span class="Number">5</span>,<span class="Number">6</span>,<span class="Number">7</span>],</span>
<span class="line"><span class="anchor" id="line-7-32"></span>    [ <span class="Number">8</span>,<span class="Number">9</span>, <span class="Number">10</span>, <span class="Number">11</span>]])</span>
<span class="line"><span class="anchor" id="line-8-29"></span>>>></span>
<span class="line"><span class="anchor" id="line-9-27"></span>>>> <span class="ID">a</span>[<span class="ID">b1</span>]                                  <span class="Comment"># same thing</span></span>
<span class="line"><span class="anchor" id="line-10-24"></span><span class="ID">array</span>([[ <span class="Number">4</span>,<span class="Number">5</span>,<span class="Number">6</span>,<span class="Number">7</span>],</span>
<span class="line"><span class="anchor" id="line-11-20"></span>    [ <span class="Number">8</span>,<span class="Number">9</span>, <span class="Number">10</span>, <span class="Number">11</span>]])</span>
<span class="line"><span class="anchor" id="line-12-18"></span>>>></span>
<span class="line"><span class="anchor" id="line-13-18"></span>>>> <span class="ID">a</span>[:,<span class="ID">b2</span>]                               <span class="Comment"># selecting columns</span></span>
<span class="line"><span class="anchor" id="line-14-16"></span><span class="ID">array</span>([[ <span class="Number">0</span>,<span class="Number">2</span>],</span>
<span class="line"><span class="anchor" id="line-15-13"></span>    [ <span class="Number">4</span>,<span class="Number">6</span>],</span>
<span class="line"><span class="anchor" id="line-16-12"></span>    [ <span class="Number">8</span>, <span class="Number">10</span>]])</span>
<span class="line"><span class="anchor" id="line-17-6"></span>>>></span>
<span class="line"><span class="anchor" id="line-18-6"></span>>>> <span class="ID">a</span>[<span class="ID">b1</span>,<span class="ID">b2</span>]                               <span class="Comment"># a weird thing to do</span></span>
<span class="line"><span class="anchor" id="line-19-6"></span><span class="ID">array</span>([ <span class="Number">4</span>, <span class="Number">10</span>])</span>
</pre></div></div><span class="anchor" id="line-881"></span><p class="line862">Note that the length of the 1D boolean array must coincide with the length of the dimension (or axis) you want to slice. In the previous example, <tt class="backtick">b1</tt> is a 1-rank array with length 3 (the number of <em>rows</em> in <tt class="backtick">a</tt>), and <tt class="backtick">b2</tt> (of length 4) is suitable to index the 2nd rank (columns) of <tt class="backtick">a</tt>. <span class="anchor" id="line-882"></span><span class="anchor" id="line-883"></span></p><p class="line867">
</p><h4 id="The_ix_.28.29_function">The ix_() function</h4>
<span class="anchor" id="line-884"></span><p class="line874">The ix_ function can be used to combine different vectors so as to obtain the result for each n-uplet. For example, if you want to compute all the a+b*c for all the triplets taken from each of the vectors a, b and c: <span class="anchor" id="line-885"></span><span class="anchor" id="line-886"></span></p><p class="line867"><span class="anchor" id="line-887"></span><span class="anchor" id="line-888"></span><span class="anchor" id="line-889"></span><span class="anchor" id="line-890"></span><span class="anchor" id="line-891"></span><span class="anchor" id="line-892"></span><span class="anchor" id="line-893"></span><span class="anchor" id="line-894"></span><span class="anchor" id="line-895"></span><span class="anchor" id="line-896"></span><span class="anchor" id="line-897"></span><span class="anchor" id="line-898"></span><span class="anchor" id="line-899"></span><span class="anchor" id="line-900"></span><span class="anchor" id="line-901"></span><span class="anchor" id="line-902"></span><span class="anchor" id="line-903"></span><span class="anchor" id="line-904"></span><span class="anchor" id="line-905"></span><span class="anchor" id="line-906"></span><span class="anchor" id="line-907"></span><span class="anchor" id="line-908"></span><span class="anchor" id="line-909"></span><span class="anchor" id="line-910"></span><span class="anchor" id="line-911"></span><span class="anchor" id="line-912"></span><span class="anchor" id="line-913"></span><span class="anchor" id="line-914"></span><span class="anchor" id="line-915"></span><span class="anchor" id="line-916"></span><span class="anchor" id="line-917"></span><span class="anchor" id="line-918"></span><span class="anchor" id="line-919"></span><span class="anchor" id="line-920"></span><span class="anchor" id="line-921"></span><span class="anchor" id="line-922"></span><span class="anchor" id="line-923"></span><span class="anchor" id="line-924"></span><span class="anchor" id="line-925"></span><span class="anchor" id="line-926"></span><span class="anchor" id="line-1-99"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-c7a4ed917842968cc4da6ec233409907d943d18b" lang="en"><span class="line"><span class="anchor" id="line-1-100"></span>>>> <span class="ID">a</span> = <span class="ID">array</span>([<span class="Number">2</span>,<span class="Number">3</span>,<span class="Number">4</span>,<span class="Number">5</span>])</span>
<span class="line"><span class="anchor" id="line-2-49"></span>>>> <span class="ID">b</span> = <span class="ID">array</span>([<span class="Number">8</span>,<span class="Number">5</span>,<span class="Number">4</span>])</span>
<span class="line"><span class="anchor" id="line-3-47"></span>>>> <span class="ID">c</span> = <span class="ID">array</span>([<span class="Number">5</span>,<span class="Number">4</span>,<span class="Number">6</span>,<span class="Number">8</span>,<span class="Number">3</span>])</span>
<span class="line"><span class="anchor" id="line-4-46"></span>>>> <span class="ID">ax</span>,<span class="ID">bx</span>,<span class="ID">cx</span> = <span class="ID">ix_</span>(<span class="ID">a</span>,<span class="ID">b</span>,<span class="ID">c</span>)</span>
<span class="line"><span class="anchor" id="line-5-37"></span>>>> <span class="ID">ax</span></span>
<span class="line"><span class="anchor" id="line-6-36"></span><span class="ID">array</span>([[[<span class="Number">2</span>]],</span>
<span class="line"><span class="anchor" id="line-7-33"></span></span>
<span class="line"><span class="anchor" id="line-8-30"></span>    [[<span class="Number">3</span>]],</span>
<span class="line"><span class="anchor" id="line-9-28"></span></span>
<span class="line"><span class="anchor" id="line-10-25"></span>    [[<span class="Number">4</span>]],</span>
<span class="line"><span class="anchor" id="line-11-21"></span></span>
<span class="line"><span class="anchor" id="line-12-19"></span>    [[<span class="Number">5</span>]]])</span>
<span class="line"><span class="anchor" id="line-13-19"></span>>>> <span class="ID">bx</span></span>
<span class="line"><span class="anchor" id="line-14-17"></span><span class="ID">array</span>([[[<span class="Number">8</span>],</span>
<span class="line"><span class="anchor" id="line-15-14"></span>    [<span class="Number">5</span>],</span>
<span class="line"><span class="anchor" id="line-16-13"></span>    [<span class="Number">4</span>]]])</span>
<span class="line"><span class="anchor" id="line-17-7"></span>>>> <span class="ID">cx</span></span>
<span class="line"><span class="anchor" id="line-18-7"></span><span class="ID">array</span>([[[<span class="Number">5</span>, <span class="Number">4</span>, <span class="Number">6</span>, <span class="Number">8</span>, <span class="Number">3</span>]]])</span>
<span class="line"><span class="anchor" id="line-19-7"></span>>>> <span class="ID">ax</span>.<span class="ID">shape</span>, <span class="ID">bx</span>.<span class="ID">shape</span>, <span class="ID">cx</span>.<span class="ID">shape</span></span>
<span class="line"><span class="anchor" id="line-20-5"></span>((<span class="Number">4</span>, <span class="Number">1</span>, <span class="Number">1</span>), (<span class="Number">1</span>, <span class="Number">3</span>, <span class="Number">1</span>), (<span class="Number">1</span>, <span class="Number">1</span>, <span class="Number">5</span>))</span>
<span class="line"><span class="anchor" id="line-21-4"></span>>>> <span class="ID">result</span> = <span class="ID">ax</span>+<span class="ID">bx</span>*<span class="ID">cx</span></span>
<span class="line"><span class="anchor" id="line-22-4"></span>>>> <span class="ID">result</span></span>
<span class="line"><span class="anchor" id="line-23-4"></span><span class="ID">array</span>([[[<span class="Number">42</span>, <span class="Number">34</span>, <span class="Number">50</span>, <span class="Number">66</span>, <span class="Number">26</span>],</span>
<span class="line"><span class="anchor" id="line-24-3"></span>    [<span class="Number">27</span>, <span class="Number">22</span>, <span class="Number">32</span>, <span class="Number">42</span>, <span class="Number">17</span>],</span>
<span class="line"><span class="anchor" id="line-25-3"></span>    [<span class="Number">22</span>, <span class="Number">18</span>, <span class="Number">26</span>, <span class="Number">34</span>, <span class="Number">14</span>]],</span>
<span class="line"><span class="anchor" id="line-26-2"></span>    [[<span class="Number">43</span>, <span class="Number">35</span>, <span class="Number">51</span>, <span class="Number">67</span>, <span class="Number">27</span>],</span>
<span class="line"><span class="anchor" id="line-27-1"></span>    [<span class="Number">28</span>, <span class="Number">23</span>, <span class="Number">33</span>, <span class="Number">43</span>, <span class="Number">18</span>],</span>
<span class="line"><span class="anchor" id="line-28-1"></span>    [<span class="Number">23</span>, <span class="Number">19</span>, <span class="Number">27</span>, <span class="Number">35</span>, <span class="Number">15</span>]],</span>
<span class="line"><span class="anchor" id="line-29-1"></span>    [[<span class="Number">44</span>, <span class="Number">36</span>, <span class="Number">52</span>, <span class="Number">68</span>, <span class="Number">28</span>],</span>
<span class="line"><span class="anchor" id="line-30-1"></span>    [<span class="Number">29</span>, <span class="Number">24</span>, <span class="Number">34</span>, <span class="Number">44</span>, <span class="Number">19</span>],</span>
<span class="line"><span class="anchor" id="line-31-1"></span>    [<span class="Number">24</span>, <span class="Number">20</span>, <span class="Number">28</span>, <span class="Number">36</span>, <span class="Number">16</span>]],</span>
<span class="line"><span class="anchor" id="line-32-1"></span>    [[<span class="Number">45</span>, <span class="Number">37</span>, <span class="Number">53</span>, <span class="Number">69</span>, <span class="Number">29</span>],</span>
<span class="line"><span class="anchor" id="line-33-1"></span>    [<span class="Number">30</span>, <span class="Number">25</span>, <span class="Number">35</span>, <span class="Number">45</span>, <span class="Number">20</span>],</span>
<span class="line"><span class="anchor" id="line-34-1"></span>    [<span class="Number">25</span>, <span class="Number">21</span>, <span class="Number">29</span>, <span class="Number">37</span>, <span class="Number">17</span>]]])</span>
<span class="line"><span class="anchor" id="line-35-1"></span>>>> <span class="ID">result</span>[<span class="Number">3</span>,<span class="Number">2</span>,<span class="Number">4</span>]</span>
<span class="line"><span class="anchor" id="line-36-1"></span><span class="Number">17</span></span>
<span class="line"><span class="anchor" id="line-37-1"></span>>>> <span class="ID">a</span>[<span class="Number">3</span>]+<span class="ID">b</span>[<span class="Number">2</span>]*<span class="ID">c</span>[<span class="Number">4</span>]</span>
<span class="line"><span class="anchor" id="line-38-1"></span><span class="Number">17</span></span>
</pre></div></div><span class="anchor" id="line-927"></span><p class="line874">You could also implement the reduce as follows: <span class="anchor" id="line-928"></span><span class="anchor" id="line-929"></span></p><p class="line867"><span class="anchor" id="line-930"></span><span class="anchor" id="line-931"></span><span class="anchor" id="line-932"></span><span class="anchor" id="line-933"></span><span class="anchor" id="line-934"></span><span class="anchor" id="line-935"></span><span class="anchor" id="line-936"></span><span class="anchor" id="line-937"></span><span class="anchor" id="line-1-101"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-29d0c55228c756816d5f5e816dcc9a960410e346" lang="en"><span class="line"><span class="anchor" id="line-1-102"></span><span class="ResWord">def</span> <span class="ID">ufunc_reduce</span>(<span class="ID">ufct</span>, *<span class="ID">vectors</span>):</span>
<span class="line"><span class="anchor" id="line-2-50"></span> <span class="ID">vs</span> = <span class="ID">ix_</span>(*<span class="ID">vectors</span>)</span>
<span class="line"><span class="anchor" id="line-3-48"></span> <span class="ID">r</span> = <span class="ID">ufct</span>.<span class="ID">identity</span></span>
<span class="line"><span class="anchor" id="line-4-47"></span> <span class="ResWord">for</span> <span class="ID">v</span> <span class="ResWord">in</span> <span class="ID">vs</span>:</span>
<span class="line"><span class="anchor" id="line-5-38"></span>    <span class="ID">r</span> = <span class="ID">ufct</span>(<span class="ID">r</span>,<span class="ID">v</span>)</span>
<span class="line"><span class="anchor" id="line-6-37"></span> <span class="ResWord">return</span> <span class="ID">r</span></span>
</pre></div></div><span class="anchor" id="line-938"></span><p class="line874">and then use it as: <span class="anchor" id="line-939"></span><span class="anchor" id="line-940"></span></p><p class="line867"><span class="anchor" id="line-941"></span><span class="anchor" id="line-942"></span><span class="anchor" id="line-943"></span><span class="anchor" id="line-944"></span><span class="anchor" id="line-945"></span><span class="anchor" id="line-946"></span><span class="anchor" id="line-947"></span><span class="anchor" id="line-948"></span><span class="anchor" id="line-949"></span><span class="anchor" id="line-950"></span><span class="anchor" id="line-951"></span><span class="anchor" id="line-952"></span><span class="anchor" id="line-953"></span><span class="anchor" id="line-954"></span><span class="anchor" id="line-955"></span><span class="anchor" id="line-1-103"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-70a1114cee0aa4cf9c68d930fcda140b03f07ab2" lang="en"><span class="line"><span class="anchor" id="line-1-104"></span>>>> <span class="ID">ufunc_reduce</span>(<span class="ID">add</span>,<span class="ID">a</span>,<span class="ID">b</span>,<span class="ID">c</span>)</span>
<span class="line"><span class="anchor" id="line-2-51"></span><span class="ID">array</span>([[[<span class="Number">15</span>, <span class="Number">14</span>, <span class="Number">16</span>, <span class="Number">18</span>, <span class="Number">13</span>],</span>
<span class="line"><span class="anchor" id="line-3-49"></span>    [<span class="Number">12</span>, <span class="Number">11</span>, <span class="Number">13</span>, <span class="Number">15</span>, <span class="Number">10</span>],</span>
<span class="line"><span class="anchor" id="line-4-48"></span>    [<span class="Number">11</span>, <span class="Number">10</span>, <span class="Number">12</span>, <span class="Number">14</span>,<span class="Number">9</span>]],</span>
<span class="line"><span class="anchor" id="line-5-39"></span>    [[<span class="Number">16</span>, <span class="Number">15</span>, <span class="Number">17</span>, <span class="Number">19</span>, <span class="Number">14</span>],</span>
<span class="line"><span class="anchor" id="line-6-38"></span>    [<span class="Number">13</span>, <span class="Number">12</span>, <span class="Number">14</span>, <span class="Number">16</span>, <span class="Number">11</span>],</span>
<span class="line"><span class="anchor" id="line-7-34"></span>    [<span class="Number">12</span>, <span class="Number">11</span>, <span class="Number">13</span>, <span class="Number">15</span>, <span class="Number">10</span>]],</span>
<span class="line"><span class="anchor" id="line-8-31"></span>    [[<span class="Number">17</span>, <span class="Number">16</span>, <span class="Number">18</span>, <span class="Number">20</span>, <span class="Number">15</span>],</span>
<span class="line"><span class="anchor" id="line-9-29"></span>    [<span class="Number">14</span>, <span class="Number">13</span>, <span class="Number">15</span>, <span class="Number">17</span>, <span class="Number">12</span>],</span>
<span class="line"><span class="anchor" id="line-10-26"></span>    [<span class="Number">13</span>, <span class="Number">12</span>, <span class="Number">14</span>, <span class="Number">16</span>, <span class="Number">11</span>]],</span>
<span class="line"><span class="anchor" id="line-11-22"></span>    [[<span class="Number">18</span>, <span class="Number">17</span>, <span class="Number">19</span>, <span class="Number">21</span>, <span class="Number">16</span>],</span>
<span class="line"><span class="anchor" id="line-12-20"></span>    [<span class="Number">15</span>, <span class="Number">14</span>, <span class="Number">16</span>, <span class="Number">18</span>, <span class="Number">13</span>],</span>
<span class="line"><span class="anchor" id="line-13-20"></span>    [<span class="Number">14</span>, <span class="Number">13</span>, <span class="Number">15</span>, <span class="Number">17</span>, <span class="Number">12</span>]]])</span>
</pre></div></div><span class="anchor" id="line-956"></span><p class="line862">The advantage of this version of reduce compared to the normal ufunc.reduce is that it makes use of the <a href="https://scipy.github.io/old-wiki/pages/Tentative_NumPy_Tutorial.html#head-c43f3f81719d84f09ae2b33a22eaf50b26333db8">Broadcasting Rules</a> in order to avoid creating an argument array the size of the output times the number of vectors. <span class="anchor" id="line-957"></span><span class="anchor" id="line-958"></span></p><p class="line867">
</p><h4 id="Indexing_with_strings">Indexing with strings</h4>
<span class="anchor" id="line-959"></span><p class="line862">See <a href="https://scipy.github.io/old-wiki/pages/RecordArrays.html">RecordArrays</a>. <span class="anchor" id="line-960"></span><span class="anchor" id="line-961"></span></p><p class="line867">
</p><h2 id="Linear_Algebra">Linear Algebra</h2>
<span class="anchor" id="line-962"></span><p class="line874">Work in progress.Basic linear algebra to be included here. <span class="anchor" id="line-963"></span>
</p><h4 id="Simple_Array_Operations">Simple Array Operations</h4>
<span class="anchor" id="line-964"></span><p class="line874">See linalg.py in numpy folder for more. <span class="anchor" id="line-965"></span><span class="anchor" id="line-966"></span><span class="anchor" id="line-967"></span><span class="anchor" id="line-968"></span><span class="anchor" id="line-969"></span><span class="anchor" id="line-970"></span><span class="anchor" id="line-971"></span><span class="anchor" id="line-972"></span><span class="anchor" id="line-973"></span><span class="anchor" id="line-974"></span><span class="anchor" id="line-975"></span><span class="anchor" id="line-976"></span><span class="anchor" id="line-977"></span><span class="anchor" id="line-978"></span><span class="anchor" id="line-979"></span><span class="anchor" id="line-980"></span><span class="anchor" id="line-981"></span><span class="anchor" id="line-982"></span><span class="anchor" id="line-983"></span><span class="anchor" id="line-984"></span><span class="anchor" id="line-985"></span><span class="anchor" id="line-986"></span><span class="anchor" id="line-987"></span><span class="anchor" id="line-988"></span><span class="anchor" id="line-989"></span><span class="anchor" id="line-990"></span><span class="anchor" id="line-991"></span><span class="anchor" id="line-992"></span><span class="anchor" id="line-993"></span><span class="anchor" id="line-994"></span><span class="anchor" id="line-995"></span><span class="anchor" id="line-996"></span><span class="anchor" id="line-997"></span><span class="anchor" id="line-998"></span><span class="anchor" id="line-999"></span><span class="anchor" id="line-1000"></span><span class="anchor" id="line-1001"></span><span class="anchor" id="line-1002"></span><span class="anchor" id="line-1003"></span><span class="anchor" id="line-1004"></span><span class="anchor" id="line-1005"></span><span class="anchor" id="line-1006"></span><span class="anchor" id="line-1007"></span><span class="anchor" id="line-1008"></span><span class="anchor" id="line-1009"></span><span class="anchor" id="line-1010"></span><span class="anchor" id="line-1011"></span><span class="anchor" id="line-1012"></span><span class="anchor" id="line-1013"></span><span class="anchor" id="line-1014"></span><span class="anchor" id="line-1015"></span><span class="anchor" id="line-1-105"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-99ae9b3e34d8a61940a99888c3bddb0e447175b6" lang="en"><span class="line"><span class="anchor" id="line-1-106"></span>>>> <span class="ResWord">from</span> <span class="ID">numpy</span> <span class="ResWord">import</span> *</span>
<span class="line"><span class="anchor" id="line-2-52"></span>>>> <span class="ResWord">from</span> <span class="ID">numpy.linalg</span> <span class="ResWord">import</span> *</span>
<span class="line"><span class="anchor" id="line-3-50"></span></span>
<span class="line"><span class="anchor" id="line-4-49"></span>>>> <span class="ID">a</span> = <span class="ID">array</span>([[<span class="Number">1.0</span>, <span class="Number">2.0</span>], [<span class="Number">3.0</span>, <span class="Number">4.0</span>]])</span>
<span class="line"><span class="anchor" id="line-5-40"></span>>>> <span class="ResWord">print</span> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-6-39"></span>[[ <span class="Number">1.</span><span class="Number">2.</span>]</span>
<span class="line"><span class="anchor" id="line-7-35"></span> [ <span class="Number">3.</span><span class="Number">4.</span>]]</span>
<span class="line"><span class="anchor" id="line-8-32"></span></span>
<span class="line"><span class="anchor" id="line-9-30"></span>>>> <span class="ID">a</span>.<span class="ID">transpose</span>()</span>
<span class="line"><span class="anchor" id="line-10-27"></span><span class="ID">array</span>([[ <span class="Number">1.</span>,<span class="Number">3.</span>],</span>
<span class="line"><span class="anchor" id="line-11-23"></span>    [ <span class="Number">2.</span>,<span class="Number">4.</span>]])</span>
<span class="line"><span class="anchor" id="line-12-21"></span></span>
<span class="line"><span class="anchor" id="line-13-21"></span>>>> <span class="ID">inv</span>(<span class="ID">a</span>)</span>
<span class="line"><span class="anchor" id="line-14-18"></span><span class="ID">array</span>([[-<span class="Number">2.</span> ,<span class="Number">1.</span> ],</span>
<span class="line"><span class="anchor" id="line-15-15"></span>    [ <span class="Number">1.5</span>, -<span class="Number">0.5</span>]])</span>
<span class="line"><span class="anchor" id="line-16-14"></span></span>
<span class="line"><span class="anchor" id="line-17-8"></span>>>> <span class="ID">u</span> = <span class="ID">eye</span>(<span class="Number">2</span>) <span class="Comment"># unit 2x2 matrix; "eye" represents "I"</span></span>
<span class="line"><span class="anchor" id="line-18-8"></span>>>> <span class="ID">u</span></span>
<span class="line"><span class="anchor" id="line-19-8"></span><span class="ID">array</span>([[ <span class="Number">1.</span>,<span class="Number">0.</span>],</span>
<span class="line"><span class="anchor" id="line-20-6"></span>    [ <span class="Number">0.</span>,<span class="Number">1.</span>]])</span>
<span class="line"><span class="anchor" id="line-21-5"></span>>>> <span class="ID">j</span> = <span class="ID">array</span>([[<span class="Number">0.0</span>, -<span class="Number">1.0</span>], [<span class="Number">1.0</span>, <span class="Number">0.0</span>]])</span>
<span class="line"><span class="anchor" id="line-22-5"></span></span>
<span class="line"><span class="anchor" id="line-23-5"></span>>>> <span class="ID">dot</span> (<span class="ID">j</span>, <span class="ID">j</span>) <span class="Comment"># matrix product</span></span>
<span class="line"><span class="anchor" id="line-24-4"></span><span class="ID">array</span>([[-<span class="Number">1.</span>,<span class="Number">0.</span>],</span>
<span class="line"><span class="anchor" id="line-25-4"></span>    [ <span class="Number">0.</span>, -<span class="Number">1.</span>]])</span>
<span class="line"><span class="anchor" id="line-26-3"></span></span>
<span class="line"><span class="anchor" id="line-27-2"></span>>>> <span class="ID">trace</span>(<span class="ID">u</span>)<span class="Comment"># trace</span></span>
<span class="line"><span class="anchor" id="line-28-2"></span><span class="Number">2.0</span></span>
<span class="line"><span class="anchor" id="line-29-2"></span></span>
<span class="line"><span class="anchor" id="line-30-2"></span>>>> <span class="ID">y</span> = <span class="ID">array</span>([[<span class="Number">5.</span>], [<span class="Number">7.</span>]])</span>
<span class="line"><span class="anchor" id="line-31-2"></span>>>> <span class="ID">solve</span>(<span class="ID">a</span>, <span class="ID">y</span>)</span>
<span class="line"><span class="anchor" id="line-32-2"></span><span class="ID">array</span>([[-<span class="Number">3.</span>],</span>
<span class="line"><span class="anchor" id="line-33-2"></span>    [ <span class="Number">4.</span>]])</span>
<span class="line"><span class="anchor" id="line-34-2"></span></span>
<span class="line"><span class="anchor" id="line-35-2"></span>>>> <span class="ID">eig</span>(<span class="ID">j</span>)</span>
<span class="line"><span class="anchor" id="line-36-2"></span>(<span class="ID">array</span>([ <span class="Number">0.</span>+<span class="Number">1.j</span>,<span class="Number">0.</span>-<span class="Number">1.j</span>]),</span>
<span class="line"><span class="anchor" id="line-37-2"></span><span class="ID">array</span>([[ <span class="Number">0.70710678</span>+<span class="Number">0.j</span>,<span class="Number">0.70710678</span>+<span class="Number">0.j</span>],</span>
<span class="line"><span class="anchor" id="line-38-2"></span>    [ <span class="Number">0.00000000</span>-<span class="Number">0.70710678j</span>,<span class="Number">0.00000000</span>+<span class="Number">0.70710678j</span>]]))</span>
<span class="line"><span class="anchor" id="line-39-1"></span><span class="ID">Parameters</span>:</span>
<span class="line"><span class="anchor" id="line-40-1"></span> <span class="ID">square</span> <span class="ID">matrix</span></span>
<span class="line"><span class="anchor" id="line-41-1"></span></span>
<span class="line"><span class="anchor" id="line-42-1"></span><span class="ID">Returns</span></span>
<span class="line"><span class="anchor" id="line-43-1"></span> <span class="ID">The</span> <span class="ID">eigenvalues</span>, <span class="ID">each</span> <span class="ID">repeated</span> <span class="ID">according</span> <span class="ID">to</span> <span class="ID">its</span> <span class="ID">multiplicity</span>.</span>
<span class="line"><span class="anchor" id="line-44-1"></span></span>
<span class="line"><span class="anchor" id="line-45-1"></span> <span class="ID">The</span> <span class="ID">normalized</span> (<span class="ID">unit</span> <span class="String">"</span><span class="String">length</span><span class="String">"</span>) <span class="ID">eigenvectors</span>, <span class="ID">such</span> <span class="ID">that</span> <span class="ID">the</span></span>
<span class="line"><span class="anchor" id="line-46-1"></span> <span class="ID">column</span> <span class="String">``</span><span class="ID">v</span>[:,<span class="ID">i</span>]<span class="String">``</span> <span class="ResWord">is</span> <span class="ID">the</span> <span class="ID">eigenvector</span> <span class="ID">corresponding</span> <span class="ID">to</span> <span class="ID">the</span></span>
<span class="line"><span class="anchor" id="line-47-1"></span> <span class="ID">eigenvalue</span> <span class="String">``</span><span class="ID">w</span>[<span class="ID">i</span>]<span class="String">``</span> .</span>
</pre></div></div><span class="anchor" id="line-1016"></span><span class="anchor" id="line-1017"></span><p class="line867">
</p><h4 id="The_Matrix_Class">The Matrix Class</h4>
<span class="anchor" id="line-1018"></span><p class="line874">Here is a short intro to the Matrix class. <span class="anchor" id="line-1019"></span><span class="anchor" id="line-1020"></span></p><p class="line867"><span class="anchor" id="line-1021"></span><span class="anchor" id="line-1022"></span><span class="anchor" id="line-1023"></span><span class="anchor" id="line-1024"></span><span class="anchor" id="line-1025"></span><span class="anchor" id="line-1026"></span><span class="anchor" id="line-1027"></span><span class="anchor" id="line-1028"></span><span class="anchor" id="line-1029"></span><span class="anchor" id="line-1030"></span><span class="anchor" id="line-1031"></span><span class="anchor" id="line-1032"></span><span class="anchor" id="line-1033"></span><span class="anchor" id="line-1034"></span><span class="anchor" id="line-1035"></span><span class="anchor" id="line-1036"></span><span class="anchor" id="line-1037"></span><span class="anchor" id="line-1038"></span><span class="anchor" id="line-1039"></span><span class="anchor" id="line-1040"></span><span class="anchor" id="line-1041"></span><span class="anchor" id="line-1042"></span><span class="anchor" id="line-1043"></span><span class="anchor" id="line-1044"></span><span class="anchor" id="line-1045"></span><span class="anchor" id="line-1046"></span><span class="anchor" id="line-1047"></span><span class="anchor" id="line-1048"></span><span class="anchor" id="line-1049"></span><span class="anchor" id="line-1050"></span><span class="anchor" id="line-1051"></span><span class="anchor" id="line-1-107"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-dcafd585ba3f6fec1a673f7aa227888ac143a416" lang="en"><span class="line"><span class="anchor" id="line-1-108"></span>>>> <span class="ID">A</span> = <span class="ID">matrix</span>(<span class="String">'</span><span class="String">1.0 2.0; 3.0 4.0</span><span class="String">'</span>)</span>
<span class="line"><span class="anchor" id="line-2-53"></span>>>> <span class="ID">A</span></span>
<span class="line"><span class="anchor" id="line-3-51"></span>[[ <span class="Number">1.</span><span class="Number">2.</span>]</span>
<span class="line"><span class="anchor" id="line-4-50"></span> [ <span class="Number">3.</span><span class="Number">4.</span>]]</span>
<span class="line"><span class="anchor" id="line-5-41"></span>>>> <span class="ResWord">type</span>(<span class="ID">A</span>)<span class="Comment"># file where class is defined</span></span>
<span class="line"><span class="anchor" id="line-6-40"></span><<span class="ResWord">class</span> '<span class="ID">numpy</span>.<span class="ID">matrixlib</span>.<span class="ID">defmatrix</span>.<span class="ID">matrix</span><span class="String">'</span><span class="String">></span></span>
<span class="line"><span class="anchor" id="line-7-36"></span></span>
<span class="line"><span class="anchor" id="line-8-33"></span>>>> <span class="ID">A</span>.<span class="ID">T</span><span class="Comment"># transpose</span></span>
<span class="line"><span class="anchor" id="line-9-31"></span>[[ <span class="Number">1.</span><span class="Number">3.</span>]</span>
<span class="line"><span class="anchor" id="line-10-28"></span> [ <span class="Number">2.</span><span class="Number">4.</span>]]</span>
<span class="line"><span class="anchor" id="line-11-24"></span></span>
<span class="line"><span class="anchor" id="line-12-22"></span>>>> <span class="ID">X</span> = <span class="ID">matrix</span>(<span class="String">'</span><span class="String">5.0 7.0</span><span class="String">'</span>)</span>
<span class="line"><span class="anchor" id="line-13-22"></span>>>> <span class="ID">Y</span> = <span class="ID">X</span>.<span class="ID">T</span></span>
<span class="line"><span class="anchor" id="line-14-19"></span>>>> <span class="ID">Y</span></span>
<span class="line"><span class="anchor" id="line-15-16"></span>[[<span class="Number">5.</span>]</span>
<span class="line"><span class="anchor" id="line-16-15"></span> [<span class="Number">7.</span>]]</span>
<span class="line"><span class="anchor" id="line-17-9"></span></span>
<span class="line"><span class="anchor" id="line-18-9"></span>>>> <span class="ResWord">print</span> <span class="ID">A</span>*<span class="ID">Y</span><span class="Comment"># matrix multiplication</span></span>
<span class="line"><span class="anchor" id="line-19-9"></span>[[<span class="Number">19.</span>]</span>
<span class="line"><span class="anchor" id="line-20-7"></span> [<span class="Number">43.</span>]]</span>
<span class="line"><span class="anchor" id="line-21-6"></span></span>
<span class="line"><span class="anchor" id="line-22-6"></span>>>> <span class="ResWord">print</span> <span class="ID">A</span>.<span class="ID">I</span><span class="Comment"># inverse</span></span>
<span class="line"><span class="anchor" id="line-23-6"></span>[[-<span class="Number">2.</span> <span class="Number">1.</span> ]</span>
<span class="line"><span class="anchor" id="line-24-5"></span> [ <span class="Number">1.5</span> -<span class="Number">0.5</span>]]</span>
<span class="line"><span class="anchor" id="line-25-5"></span></span>
<span class="line"><span class="anchor" id="line-26-4"></span>>>> <span class="ID">solve</span>(<span class="ID">A</span>, <span class="ID">Y</span>)<span class="Comment"># solving linear equation</span></span>
<span class="line"><span class="anchor" id="line-27-3"></span><span class="ID">matrix</span>([[-<span class="Number">3.</span>],</span>
<span class="line"><span class="anchor" id="line-28-3"></span>    [ <span class="Number">4.</span>]])</span>
</pre></div></div><span class="anchor" id="line-1052"></span><p class="line867">
</p><h4 id="Indexing:_Comparing_Matrices_and_2D_Arrays">Indexing: Comparing Matrices and 2D Arrays</h4>
<span class="anchor" id="line-1053"></span><p class="line862">Note that there are some important differences between <a class="nonexistent" href="https://scipy.github.io/old-wiki/pages/NumPy.html">NumPy</a> arrays and matrices. <a class="nonexistent" href="https://scipy.github.io/old-wiki/pages/NumPy.html">NumPy</a> provides two fundamental objects: an N-dimensional array object and a universal function object. Other objects are built on top of these. In particular, matrices are 2-dimensional array objects that inherit from the <a class="nonexistent" href="https://scipy.github.io/old-wiki/pages/NumPy.html">NumPy</a> array object. For both arrays and matrices, indices must consist of a proper combination of one or more of the following: integer scalars, ellipses, a list of integers or boolean values, a tuple of integers or boolean values, and a 1-dimensional array of integer or boolean values. A matrix can be used as an index for matrices, but commonly an array, list, or other form is needed to accomplish a given task. <span class="anchor" id="line-1054"></span><span class="anchor" id="line-1055"></span></p><p class="line874">As usual in Python, indexing is zero-based. Traditionally we represent a 2D array or matrix as a rectangular array of rows and columns, where movement along axis 0 is movement across rows, while movement along axis 1 is movement across columns. <span class="anchor" id="line-1056"></span><span class="anchor" id="line-1057"></span></p><p class="line874">Let's make an array and matrix to slice: <span class="anchor" id="line-1058"></span><span class="anchor" id="line-1059"></span></p><p class="line867"><span class="anchor" id="line-1060"></span><span class="anchor" id="line-1061"></span><span class="anchor" id="line-1062"></span><span class="anchor" id="line-1063"></span><span class="anchor" id="line-1064"></span><span class="anchor" id="line-1065"></span><span class="anchor" id="line-1066"></span><span class="anchor" id="line-1067"></span><span class="anchor" id="line-1068"></span><span class="anchor" id="line-1069"></span><span class="anchor" id="line-1070"></span><span class="anchor" id="line-1071"></span><span class="anchor" id="line-1072"></span><span class="anchor" id="line-1073"></span><span class="anchor" id="line-1074"></span><span class="anchor" id="line-1075"></span><span class="anchor" id="line-1076"></span><span class="anchor" id="line-1-109"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-da4a723ac957df43d883c991c0ba27f274ccc625" lang="en"><span class="line"><span class="anchor" id="line-1-110"></span>>>> <span class="ID">A</span> = <span class="ID">arange</span>(<span class="Number">12</span>)</span>
<span class="line"><span class="anchor" id="line-2-54"></span>>>> <span class="ID">A</span></span>
<span class="line"><span class="anchor" id="line-3-52"></span><span class="ID">array</span>([ <span class="Number">0</span>,<span class="Number">1</span>,<span class="Number">2</span>,<span class="Number">3</span>,<span class="Number">4</span>,<span class="Number">5</span>,<span class="Number">6</span>,<span class="Number">7</span>,<span class="Number">8</span>,<span class="Number">9</span>, <span class="Number">10</span>, <span class="Number">11</span>])</span>
<span class="line"><span class="anchor" id="line-4-51"></span>>>> <span class="ID">A</span>.<span class="ID">shape</span> = (<span class="Number">3</span>,<span class="Number">4</span>)</span>
<span class="line"><span class="anchor" id="line-5-42"></span>>>> <span class="ID">M</span> = <span class="ID">mat</span>(<span class="ID">A</span>.<span class="ID">copy</span>())</span>
<span class="line"><span class="anchor" id="line-6-41"></span>>>> <span class="ResWord">print</span> <span class="ResWord">type</span>(<span class="ID">A</span>),<span class="String">"</span><span class="String"></span><span class="String">"</span>,<span class="ResWord">type</span>(<span class="ID">M</span>)</span>
<span class="line"><span class="anchor" id="line-7-37"></span><<span class="ResWord">type</span> <span class="String">'</span><span class="String">numpy.ndarray</span><span class="String">'</span>> <<span class="ResWord">class</span> '<span class="ID">numpy</span>.<span class="ID">core</span>.<span class="ID">defmatrix</span>.<span class="ID">matrix</span><span class="String">'</span><span class="String">></span></span>
<span class="line"><span class="anchor" id="line-8-34"></span>>>> <span class="ResWord">print</span> <span class="ID">A</span></span>
<span class="line"><span class="anchor" id="line-9-32"></span>[[ <span class="Number">0</span><span class="Number">1</span><span class="Number">2</span><span class="Number">3</span>]</span>
<span class="line"><span class="anchor" id="line-10-29"></span> [ <span class="Number">4</span><span class="Number">5</span><span class="Number">6</span><span class="Number">7</span>]</span>
<span class="line"><span class="anchor" id="line-11-25"></span> [ <span class="Number">8</span><span class="Number">9</span> <span class="Number">10</span> <span class="Number">11</span>]]</span>
<span class="line"><span class="anchor" id="line-12-23"></span>>>> <span class="ResWord">print</span> <span class="ID">M</span></span>
<span class="line"><span class="anchor" id="line-13-23"></span>[[ <span class="Number">0</span><span class="Number">1</span><span class="Number">2</span><span class="Number">3</span>]</span>
<span class="line"><span class="anchor" id="line-14-20"></span> [ <span class="Number">4</span><span class="Number">5</span><span class="Number">6</span><span class="Number">7</span>]</span>
<span class="line"><span class="anchor" id="line-15-17"></span> [ <span class="Number">8</span><span class="Number">9</span> <span class="Number">10</span> <span class="Number">11</span>]]</span>
</pre></div></div><span class="anchor" id="line-1077"></span><p class="line862">Now, let's take some simple slices. Basic slicing uses slice objects or integers. For example, the evaluation of A[:] and M[:] will appear familiar from Python indexing, however it is important to note that slicing <a class="nonexistent" href="https://scipy.github.io/old-wiki/pages/NumPy.html">NumPy</a> arrays does *not* make a copy of the data; slicing provides a new view of the same data. <span class="anchor" id="line-1078"></span><span class="anchor" id="line-1079"></span></p><p class="line867"><span class="anchor" id="line-1080"></span><span class="anchor" id="line-1081"></span><span class="anchor" id="line-1082"></span><span class="anchor" id="line-1083"></span><span class="anchor" id="line-1084"></span><span class="anchor" id="line-1085"></span><span class="anchor" id="line-1086"></span><span class="anchor" id="line-1087"></span><span class="anchor" id="line-1088"></span><span class="anchor" id="line-1089"></span><span class="anchor" id="line-1090"></span><span class="anchor" id="line-1091"></span><span class="anchor" id="line-1-111"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-f7e0493e494c8897179166e7ce89e37967a6aa73" lang="en"><span class="line"><span class="anchor" id="line-1-112"></span>>>> <span class="ResWord">print</span> <span class="ID">A</span>[:]; <span class="ResWord">print</span> <span class="ID">A</span>[:].<span class="ID">shape</span></span>
<span class="line"><span class="anchor" id="line-2-55"></span>[[ <span class="Number">0</span><span class="Number">1</span><span class="Number">2</span><span class="Number">3</span>]</span>
<span class="line"><span class="anchor" id="line-3-53"></span> [ <span class="Number">4</span><span class="Number">5</span><span class="Number">6</span><span class="Number">7</span>]</span>
<span class="line"><span class="anchor" id="line-4-52"></span> [ <span class="Number">8</span><span class="Number">9</span> <span class="Number">10</span> <span class="Number">11</span>]]</span>
<span class="line"><span class="anchor" id="line-5-43"></span>(<span class="Number">3</span>, <span class="Number">4</span>)</span>
<span class="line"><span class="anchor" id="line-6-42"></span>>>> <span class="ResWord">print</span> <span class="ID">M</span>[:]; <span class="ResWord">print</span> <span class="ID">M</span>[:].<span class="ID">shape</span></span>
<span class="line"><span class="anchor" id="line-7-38"></span>[[ <span class="Number">0</span><span class="Number">1</span><span class="Number">2</span><span class="Number">3</span>]</span>
<span class="line"><span class="anchor" id="line-8-35"></span> [ <span class="Number">4</span><span class="Number">5</span><span class="Number">6</span><span class="Number">7</span>]</span>
<span class="line"><span class="anchor" id="line-9-33"></span> [ <span class="Number">8</span><span class="Number">9</span> <span class="Number">10</span> <span class="Number">11</span>]]</span>
<span class="line"><span class="anchor" id="line-10-30"></span>(<span class="Number">3</span>, <span class="Number">4</span>)</span>
</pre></div></div><span class="anchor" id="line-1092"></span><p class="line874">Now for something that differs from Python indexing: you may use comma-separated indices to index along multiple axes at the same time. <span class="anchor" id="line-1093"></span><span class="anchor" id="line-1094"></span></p><p class="line867"><span class="anchor" id="line-1095"></span><span class="anchor" id="line-1096"></span><span class="anchor" id="line-1097"></span><span class="anchor" id="line-1098"></span><span class="anchor" id="line-1099"></span><span class="anchor" id="line-1100"></span><span class="anchor" id="line-1101"></span><span class="anchor" id="line-1102"></span><span class="anchor" id="line-1103"></span><span class="anchor" id="line-1104"></span><span class="anchor" id="line-1-113"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-63591fc2313654a2a319773401ac8defef4c6c67" lang="en"><span class="line"><span class="anchor" id="line-1-114"></span>>>> <span class="ResWord">print</span> <span class="ID">A</span>[:,<span class="Number">1</span>]; <span class="ResWord">print</span> <span class="ID">A</span>[:,<span class="Number">1</span>].<span class="ID">shape</span></span>
<span class="line"><span class="anchor" id="line-2-56"></span>[<span class="Number">1</span> <span class="Number">5</span> <span class="Number">9</span>]</span>
<span class="line"><span class="anchor" id="line-3-54"></span>(<span class="Number">3</span>,)</span>
<span class="line"><span class="anchor" id="line-4-53"></span>>>> <span class="ResWord">print</span> <span class="ID">M</span>[:,<span class="Number">1</span>]; <span class="ResWord">print</span> <span class="ID">M</span>[:,<span class="Number">1</span>].<span class="ID">shape</span></span>
<span class="line"><span class="anchor" id="line-5-44"></span>[[<span class="Number">1</span>]</span>
<span class="line"><span class="anchor" id="line-6-43"></span> [<span class="Number">5</span>]</span>
<span class="line"><span class="anchor" id="line-7-39"></span> [<span class="Number">9</span>]]</span>
<span class="line"><span class="anchor" id="line-8-36"></span>(<span class="Number">3</span>, <span class="Number">1</span>)</span>
</pre></div></div><span class="anchor" id="line-1105"></span><p class="line874">Notice the difference in the last two results. Use of a single colon for the 2D array produces a 1-dimensional array, while for a matrix it produces a 2-dimensional matrix. A slice of a matrix will always produce a matrix. For example, a slice M produces a matrix of shape (1,4). In contrast, a slice of an array will always produce an array of the lowest possible dimension. For example, if C were a 3-dimensional array, C[...,1] produces a 2D array while C produces a 1-dimensional array. From this point on, we will show results only for the array slice if the results for the corresponding matrix slice are identical. <span class="anchor" id="line-1106"></span><span class="anchor" id="line-1107"></span></p><p class="line874">Lets say that we wanted the 1st and 3rd column of an array. One way is to slice using a list: <span class="anchor" id="line-1108"></span><span class="anchor" id="line-1109"></span></p><p class="line867"><span class="anchor" id="line-1110"></span><span class="anchor" id="line-1111"></span><span class="anchor" id="line-1112"></span><span class="anchor" id="line-1113"></span><span class="anchor" id="line-1114"></span><span class="anchor" id="line-1115"></span><span class="anchor" id="line-1-115"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-65a815a62872549d4a4b167f086119cb42099499" lang="en"><span class="line"><span class="anchor" id="line-1-116"></span>>>> <span class="ID">A</span>[:,[<span class="Number">1</span>,<span class="Number">3</span>]]</span>
<span class="line"><span class="anchor" id="line-2-57"></span><span class="ID">array</span>([[ <span class="Number">1</span>,<span class="Number">3</span>],</span>
<span class="line"><span class="anchor" id="line-3-55"></span>    [ <span class="Number">5</span>,<span class="Number">7</span>],</span>
<span class="line"><span class="anchor" id="line-4-54"></span>    [ <span class="Number">9</span>, <span class="Number">11</span>]])</span>
</pre></div></div><span class="anchor" id="line-1116"></span><p class="line874">A slightly more complicated way is to use the take() method: <span class="anchor" id="line-1117"></span><span class="anchor" id="line-1118"></span></p><p class="line867"><span class="anchor" id="line-1119"></span><span class="anchor" id="line-1120"></span><span class="anchor" id="line-1121"></span><span class="anchor" id="line-1122"></span><span class="anchor" id="line-1123"></span><span class="anchor" id="line-1124"></span><span class="anchor" id="line-1-117"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-8ceab546afe2b99fa11109b44bde65b7eaf58b9e" lang="en"><span class="line"><span class="anchor" id="line-1-118"></span>>>> <span class="ID">A</span>[:,].<span class="ID">take</span>([<span class="Number">1</span>,<span class="Number">3</span>],<span class="ID">axis</span>=<span class="Number">1</span>)</span>
<span class="line"><span class="anchor" id="line-2-58"></span><span class="ID">array</span>([[ <span class="Number">1</span>,<span class="Number">3</span>],</span>
<span class="line"><span class="anchor" id="line-3-56"></span>    [ <span class="Number">5</span>,<span class="Number">7</span>],</span>
<span class="line"><span class="anchor" id="line-4-55"></span>    [ <span class="Number">9</span>, <span class="Number">11</span>]])</span>
</pre></div></div><span class="anchor" id="line-1125"></span><p class="line874">If we wanted to skip the first row, we could use: <span class="anchor" id="line-1126"></span><span class="anchor" id="line-1127"></span></p><p class="line867"><span class="anchor" id="line-1128"></span><span class="anchor" id="line-1129"></span><span class="anchor" id="line-1130"></span><span class="anchor" id="line-1131"></span><span class="anchor" id="line-1132"></span><span class="anchor" id="line-1-119"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-b80fdd00e4c8e7800c638fd35990761e5ab974fc" lang="en"><span class="line"><span class="anchor" id="line-1-120"></span>>>> <span class="ID">A</span>[<span class="Number">1</span>:,].<span class="ID">take</span>([<span class="Number">1</span>,<span class="Number">3</span>],<span class="ID">axis</span>=<span class="Number">1</span>)</span>
<span class="line"><span class="anchor" id="line-2-59"></span><span class="ID">array</span>([[ <span class="Number">5</span>,<span class="Number">7</span>],</span>
<span class="line"><span class="anchor" id="line-3-57"></span>    [ <span class="Number">9</span>, <span class="Number">11</span>]])</span>
</pre></div></div><span class="anchor" id="line-1133"></span><p class="line874">Or we could simply use A]. Yet another way to slice the above is to use a cross product: <span class="anchor" id="line-1134"></span><span class="anchor" id="line-1135"></span></p><p class="line867"><span class="anchor" id="line-1136"></span><span class="anchor" id="line-1137"></span><span class="anchor" id="line-1138"></span><span class="anchor" id="line-1139"></span><span class="anchor" id="line-1140"></span><span class="anchor" id="line-1-121"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-074b5d456a57c21acb8cc6ceb0eeac08fb79bff0" lang="en"><span class="line"><span class="anchor" id="line-1-122"></span>>>> <span class="ID">A</span>[<span class="ID">ix_</span>((<span class="Number">1</span>,<span class="Number">2</span>),(<span class="Number">1</span>,<span class="Number">3</span>))]</span>
<span class="line"><span class="anchor" id="line-2-60"></span><span class="ID">array</span>([[ <span class="Number">5</span>,<span class="Number">7</span>],</span>
<span class="line"><span class="anchor" id="line-3-58"></span>    [ <span class="Number">9</span>, <span class="Number">11</span>]])</span>
</pre></div></div><span class="anchor" id="line-1141"></span><p class="line874">For the reader's convenience, here is our array again: <span class="anchor" id="line-1142"></span><span class="anchor" id="line-1143"></span></p><p class="line867"><span class="anchor" id="line-1144"></span><span class="anchor" id="line-1145"></span><span class="anchor" id="line-1146"></span><span class="anchor" id="line-1147"></span><span class="anchor" id="line-1148"></span><span class="anchor" id="line-1149"></span><span class="anchor" id="line-1-123"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-e0f8fb211a570900582df6c039b8edc55905d181" lang="en"><span class="line"><span class="anchor" id="line-1-124"></span>>>> <span class="ResWord">print</span> <span class="ID">A</span></span>
<span class="line"><span class="anchor" id="line-2-61"></span>[[ <span class="Number">0</span><span class="Number">1</span><span class="Number">2</span><span class="Number">3</span>]</span>
<span class="line"><span class="anchor" id="line-3-59"></span> [ <span class="Number">4</span><span class="Number">5</span><span class="Number">6</span><span class="Number">7</span>]</span>
<span class="line"><span class="anchor" id="line-4-56"></span> [ <span class="Number">8</span><span class="Number">9</span> <span class="Number">10</span> <span class="Number">11</span>]]</span>
</pre></div></div><span class="anchor" id="line-1150"></span><p class="line874">Now let's do something a bit more complicated. Lets say that we want to retain all columns where the first row is greater than 1. One way is to create a boolean index: <span class="anchor" id="line-1151"></span><span class="anchor" id="line-1152"></span></p><p class="line867"><span class="anchor" id="line-1153"></span><span class="anchor" id="line-1154"></span><span class="anchor" id="line-1155"></span><span class="anchor" id="line-1156"></span><span class="anchor" id="line-1157"></span><span class="anchor" id="line-1158"></span><span class="anchor" id="line-1159"></span><span class="anchor" id="line-1160"></span><span class="anchor" id="line-1-125"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-b3253400e1e787d35a3d09212f9e4b61c44b6dde" lang="en"><span class="line"><span class="anchor" id="line-1-126"></span>>>> <span class="ID">A</span>[<span class="Number">0</span>,:]><span class="Number">1</span></span>
<span class="line"><span class="anchor" id="line-2-62"></span><span class="ID">array</span>([<span class="ResWord">False</span>, <span class="ResWord">False</span>, <span class="ResWord">True</span>, <span class="ResWord">True</span>], <span class="ID">dtype</span>=<span class="ResWord">bool</span>)</span>
<span class="line"><span class="anchor" id="line-3-60"></span>>>> <span class="ID">A</span>[:,<span class="ID">A</span>[<span class="Number">0</span>,:]><span class="Number">1</span>]</span>
<span class="line"><span class="anchor" id="line-4-57"></span><span class="ID">array</span>([[ <span class="Number">2</span>,<span class="Number">3</span>],</span>
<span class="line"><span class="anchor" id="line-5-45"></span>    [ <span class="Number">6</span>,<span class="Number">7</span>],</span>
<span class="line"><span class="anchor" id="line-6-44"></span>    [<span class="Number">10</span>, <span class="Number">11</span>]])</span>
</pre></div></div><span class="anchor" id="line-1161"></span><p class="line874">Just what we wanted! But indexing the matrix is not so convenient. <span class="anchor" id="line-1162"></span><span class="anchor" id="line-1163"></span></p><p class="line867"><span class="anchor" id="line-1164"></span><span class="anchor" id="line-1165"></span><span class="anchor" id="line-1166"></span><span class="anchor" id="line-1167"></span><span class="anchor" id="line-1168"></span><span class="anchor" id="line-1169"></span><span class="anchor" id="line-1-127"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-97dd602a5add7c7ea94e13ff527fd88f8ed6b05a" lang="en"><span class="line"><span class="anchor" id="line-1-128"></span>>>> <span class="ID">M</span>[<span class="Number">0</span>,:]><span class="Number">1</span></span>
<span class="line"><span class="anchor" id="line-2-63"></span><span class="ID">matrix</span>([[<span class="ResWord">False</span>, <span class="ResWord">False</span>, <span class="ResWord">True</span>, <span class="ResWord">True</span>]], <span class="ID">dtype</span>=<span class="ResWord">bool</span>)</span>
<span class="line"><span class="anchor" id="line-3-61"></span>>>> <span class="ID">M</span>[:,<span class="ID">M</span>[<span class="Number">0</span>,:]><span class="Number">1</span>]</span>
<span class="line"><span class="anchor" id="line-4-58"></span><span class="ID">matrix</span>([[<span class="Number">2</span>, <span class="Number">3</span>]])</span>
</pre></div></div><span class="anchor" id="line-1170"></span><p class="line874">The problem of course is that slicing the matrix slice produced a matrix. But matrices have a convenient 'A' attribute whose value is the array representation, so we can just do this instead: <span class="anchor" id="line-1171"></span><span class="anchor" id="line-1172"></span></p><p class="line867"><span class="anchor" id="line-1173"></span><span class="anchor" id="line-1174"></span><span class="anchor" id="line-1175"></span><span class="anchor" id="line-1176"></span><span class="anchor" id="line-1177"></span><span class="anchor" id="line-1178"></span><span class="anchor" id="line-1-129"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-3a851a5e8813f3ab8133d3ef0627aa06496afb16" lang="en"><span class="line"><span class="anchor" id="line-1-130"></span>>>> <span class="ID">M</span>[:,<span class="ID">M</span>.<span class="ID">A</span>[<span class="Number">0</span>,:]><span class="Number">1</span>]</span>
<span class="line"><span class="anchor" id="line-2-64"></span><span class="ID">matrix</span>([[ <span class="Number">2</span>,<span class="Number">3</span>],</span>
<span class="line"><span class="anchor" id="line-3-62"></span>    [ <span class="Number">6</span>,<span class="Number">7</span>],</span>
<span class="line"><span class="anchor" id="line-4-59"></span>    [<span class="Number">10</span>, <span class="Number">11</span>]])</span>
</pre></div></div><span class="anchor" id="line-1179"></span><p class="line874">If we wanted to conditionally slice the matrix in two directions, we must adjust our strategy slightly. Instead of <span class="anchor" id="line-1180"></span><span class="anchor" id="line-1181"></span></p><p class="line867"><span class="anchor" id="line-1182"></span><span class="anchor" id="line-1183"></span><span class="anchor" id="line-1184"></span><span class="anchor" id="line-1185"></span><span class="anchor" id="line-1186"></span><span class="anchor" id="line-1187"></span><span class="anchor" id="line-1-131"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-3dc886e5de89152da40a2ba96022962306f4e7de" lang="en"><span class="line"><span class="anchor" id="line-1-132"></span>>>> <span class="ID">A</span>[<span class="ID">A</span>[:,<span class="Number">0</span>]><span class="Number">2</span>,<span class="ID">A</span>[<span class="Number">0</span>,:]><span class="Number">1</span>]</span>
<span class="line"><span class="anchor" id="line-2-65"></span><span class="ID">array</span>([ <span class="Number">6</span>, <span class="Number">11</span>])</span>
<span class="line"><span class="anchor" id="line-3-63"></span>>>> <span class="ID">M</span>[<span class="ID">M</span>.<span class="ID">A</span>[:,<span class="Number">0</span>]><span class="Number">2</span>,<span class="ID">M</span>.<span class="ID">A</span>[<span class="Number">0</span>,:]><span class="Number">1</span>]</span>
<span class="line"><span class="anchor" id="line-4-60"></span><span class="ID">matrix</span>([[ <span class="Number">6</span>, <span class="Number">11</span>]])</span>
</pre></div></div><span class="anchor" id="line-1188"></span><p class="line874">we need to use the cross product 'ix_': <span class="anchor" id="line-1189"></span><span class="anchor" id="line-1190"></span></p><p class="line867"><span class="anchor" id="line-1191"></span><span class="anchor" id="line-1192"></span><span class="anchor" id="line-1193"></span><span class="anchor" id="line-1194"></span><span class="anchor" id="line-1195"></span><span class="anchor" id="line-1196"></span><span class="anchor" id="line-1197"></span><span class="anchor" id="line-1198"></span><span class="anchor" id="line-1-133"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-999b1d851faa66d579955a285af68eb01cb66ab6" lang="en"><span class="line"><span class="anchor" id="line-1-134"></span>>>> <span class="ID">A</span>[<span class="ID">numpy</span>.<span class="ID">ix_</span>(<span class="ID">A</span>[:,<span class="Number">0</span>]><span class="Number">2</span>,<span class="ID">A</span>[<span class="Number">0</span>,:]><span class="Number">1</span>)]</span>
<span class="line"><span class="anchor" id="line-2-66"></span><span class="ID">array</span>([[ <span class="Number">6</span>,<span class="Number">7</span>],</span>
<span class="line"><span class="anchor" id="line-3-64"></span>    [<span class="Number">10</span>, <span class="Number">11</span>]])</span>
<span class="line"><span class="anchor" id="line-4-61"></span>>>> <span class="ID">M</span>[<span class="ID">numpy</span>.<span class="ID">ix_</span>(<span class="ID">M</span>.<span class="ID">A</span>[:,<span class="Number">0</span>]><span class="Number">2</span>,<span class="ID">M</span>.<span class="ID">A</span>[<span class="Number">0</span>,:]><span class="Number">1</span>)]</span>
<span class="line"><span class="anchor" id="line-5-46"></span><span class="ID">matrix</span>([[ <span class="Number">6</span>,<span class="Number">7</span>],</span>
<span class="line"><span class="anchor" id="line-6-45"></span>    [<span class="Number">10</span>, <span class="Number">11</span>]])</span>
</pre></div></div><span class="anchor" id="line-1199"></span><span class="anchor" id="line-1200"></span><p class="line867">
</p><h2 id="Tricks_and_Tips">Tricks and Tips</h2>
<span class="anchor" id="line-1201"></span><p class="line874">Here we give a list of short and useful tips. <span class="anchor" id="line-1202"></span><span class="anchor" id="line-1203"></span></p><p class="line867">
</p><h4 id="A.22Automatic.22_Reshaping">"Automatic" Reshaping</h4>
<span class="anchor" id="line-1204"></span><p class="line874">To change the dimensions of an array, you can omit one of the sizes which will then be deduced automatically: <span class="anchor" id="line-1205"></span><span class="anchor" id="line-1206"></span></p><p class="line867"><span class="anchor" id="line-1207"></span><span class="anchor" id="line-1208"></span><span class="anchor" id="line-1209"></span><span class="anchor" id="line-1210"></span><span class="anchor" id="line-1211"></span><span class="anchor" id="line-1212"></span><span class="anchor" id="line-1213"></span><span class="anchor" id="line-1214"></span><span class="anchor" id="line-1215"></span><span class="anchor" id="line-1216"></span><span class="anchor" id="line-1217"></span><span class="anchor" id="line-1218"></span><span class="anchor" id="line-1219"></span><span class="anchor" id="line-1220"></span><span class="anchor" id="line-1221"></span><span class="anchor" id="line-1222"></span><span class="anchor" id="line-1223"></span><span class="anchor" id="line-1-135"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-37bfd3c819e28013dac0397e46926eeb1b514d30" lang="en"><span class="line"><span class="anchor" id="line-1-136"></span>>>> <span class="ID">a</span> = <span class="ID">arange</span>(<span class="Number">30</span>)</span>
<span class="line"><span class="anchor" id="line-2-67"></span>>>> <span class="ID">a</span>.<span class="ID">shape</span> = <span class="Number">2</span>,-<span class="Number">1</span>,<span class="Number">3</span><span class="Comment"># -1 means "whatever is needed"</span></span>
<span class="line"><span class="anchor" id="line-3-65"></span>>>> <span class="ID">a</span>.<span class="ID">shape</span></span>
<span class="line"><span class="anchor" id="line-4-62"></span>(<span class="Number">2</span>, <span class="Number">5</span>, <span class="Number">3</span>)</span>
<span class="line"><span class="anchor" id="line-5-47"></span>>>> <span class="ID">a</span></span>
<span class="line"><span class="anchor" id="line-6-46"></span><span class="ID">array</span>([[[ <span class="Number">0</span>,<span class="Number">1</span>,<span class="Number">2</span>],</span>
<span class="line"><span class="anchor" id="line-7-40"></span>    [ <span class="Number">3</span>,<span class="Number">4</span>,<span class="Number">5</span>],</span>
<span class="line"><span class="anchor" id="line-8-37"></span>    [ <span class="Number">6</span>,<span class="Number">7</span>,<span class="Number">8</span>],</span>
<span class="line"><span class="anchor" id="line-9-34"></span>    [ <span class="Number">9</span>, <span class="Number">10</span>, <span class="Number">11</span>],</span>
<span class="line"><span class="anchor" id="line-10-31"></span>    [<span class="Number">12</span>, <span class="Number">13</span>, <span class="Number">14</span>]],</span>
<span class="line"><span class="anchor" id="line-11-26"></span>    [[<span class="Number">15</span>, <span class="Number">16</span>, <span class="Number">17</span>],</span>
<span class="line"><span class="anchor" id="line-12-24"></span>    [<span class="Number">18</span>, <span class="Number">19</span>, <span class="Number">20</span>],</span>
<span class="line"><span class="anchor" id="line-13-24"></span>    [<span class="Number">21</span>, <span class="Number">22</span>, <span class="Number">23</span>],</span>
<span class="line"><span class="anchor" id="line-14-21"></span>    [<span class="Number">24</span>, <span class="Number">25</span>, <span class="Number">26</span>],</span>
<span class="line"><span class="anchor" id="line-15-18"></span>    [<span class="Number">27</span>, <span class="Number">28</span>, <span class="Number">29</span>]]])</span>
</pre></div></div><span class="anchor" id="line-1224"></span><p class="line867">
</p><h4 id="Vector_Stacking">Vector Stacking</h4>
<span class="anchor" id="line-1225"></span><p class="line862">How do we construct a 2D array from a list of equally-sized row vectors? In MATLAB this is quite easy: if <tt class="backtick">x</tt> and <tt class="backtick">y</tt> are two vectors of the same length you only need do <tt class="backtick">m=</tt>. In NumPy this works via the functions <tt class="backtick">column_stack</tt>, <tt class="backtick">dstack</tt>, <tt class="backtick">hstack</tt> and <tt class="backtick">vstack</tt>, depending on the dimension in which the stacking is to be done. For example: <span class="anchor" id="line-1226"></span><span class="anchor" id="line-1227"></span></p><p class="line867"><span class="anchor" id="line-1228"></span><span class="anchor" id="line-1229"></span><span class="anchor" id="line-1230"></span><span class="anchor" id="line-1231"></span><span class="anchor" id="line-1232"></span><span class="anchor" id="line-1233"></span><span class="anchor" id="line-1234"></span><span class="anchor" id="line-1-137"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-9bb5ed59afbeeb9bc7c16fc5b493a235cb07dbd3" lang="en"><span class="line"><span class="anchor" id="line-1-138"></span><span class="ID">x</span> = <span class="ID">arange</span>(<span class="Number">0</span>,<span class="Number">10</span>,<span class="Number">2</span>)                   <span class="Comment"># x=()</span></span>
<span class="line"><span class="anchor" id="line-2-68"></span><span class="ID">y</span> = <span class="ID">arange</span>(<span class="Number">5</span>)                      <span class="Comment"># y=()</span></span>
<span class="line"><span class="anchor" id="line-3-66"></span><span class="ID">m</span> = <span class="ID">vstack</span>([<span class="ID">x</span>,<span class="ID">y</span>])                   <span class="Comment"># m=([,</span></span>
<span class="line"><span class="anchor" id="line-4-63"></span>                                     <span class="Comment"># ])</span></span>
<span class="line"><span class="anchor" id="line-5-48"></span><span class="ID">xy</span> = <span class="ID">hstack</span>([<span class="ID">x</span>,<span class="ID">y</span>])                   <span class="Comment"># xy =()</span></span>
</pre></div></div><span class="anchor" id="line-1235"></span><p class="line874">The logic behind those functions in more than two dimensions can be strange. <span class="anchor" id="line-1236"></span><span class="anchor" id="line-1237"></span></p><p class="line862">See also <a href="https://scipy.github.io/old-wiki/pages/NumPy_for_Matlab_Users.html">NumPy_for_Matlab_Users</a> and add your new findings there.<span class="anchor" id="line-1238"></span><span class="anchor" id="line-1239"></span></p><p class="line867">
</p><h4 id="Histograms">Histograms</h4>
<span class="anchor" id="line-1240"></span><p class="line862">The NumPy <tt class="backtick">histogram</tt> function applied to an array returns a pair of vectors: the histogram of the array and the vector of bins. Beware: <tt class="backtick">matplotlib</tt> also has a function to build histograms (called <tt class="backtick">hist</tt>, as in Matlab) that differs from the one in NumPy. The main difference is that <tt class="backtick">pylab.hist</tt> plots the histogram automatically, while <tt class="backtick">numpy.histogram</tt> only generates the data. <span class="anchor" id="line-1241"></span><span class="anchor" id="line-1242"></span></p><p class="line867"><span class="anchor" id="line-1243"></span><span class="anchor" id="line-1244"></span><span class="anchor" id="line-1245"></span><span class="anchor" id="line-1246"></span><span class="anchor" id="line-1247"></span><span class="anchor" id="line-1248"></span><span class="anchor" id="line-1249"></span><span class="anchor" id="line-1250"></span><span class="anchor" id="line-1251"></span><span class="anchor" id="line-1252"></span><span class="anchor" id="line-1253"></span><span class="anchor" id="line-1254"></span><span class="anchor" id="line-1255"></span><span class="anchor" id="line-1256"></span><span class="anchor" id="line-1-139"></span></p><div class="highlight python"><div class="codearea" dir="ltr" lang="en"><pre dir="ltr" id="CA-a948509aa9faa662385e91258a82cae51efd9081" lang="en"><span class="line"><span class="anchor" id="line-1-140"></span><span class="ResWord">import</span> <span class="ID">numpy</span></span>
<span class="line"><span class="anchor" id="line-2-69"></span><span class="ResWord">import</span> <span class="ID">pylab</span></span>
<span class="line"><span class="anchor" id="line-3-67"></span><span class="Comment"># Build a vector of 10000 normal deviates with variance 0.5^2 and mean 2</span></span>
<span class="line"><span class="anchor" id="line-4-64"></span><span class="ID">mu</span>, <span class="ID">sigma</span> = <span class="Number">2</span>, <span class="Number">0.5</span></span>
<span class="line"><span class="anchor" id="line-5-49"></span><span class="ID">v</span> = <span class="ID">numpy</span>.<span class="ID">random</span>.<span class="ID">normal</span>(<span class="ID">mu</span>,<span class="ID">sigma</span>,<span class="Number">10000</span>)</span>
<span class="line"><span class="anchor" id="line-6-47"></span><span class="Comment"># Plot a normalized histogram with 50 bins</span></span>
<span class="line"><span class="anchor" id="line-7-41"></span><span class="ID">pylab</span>.<span class="ID">hist</span>(<span class="ID">v</span>, <span class="ID">bins</span>=<span class="Number">50</span>, <span class="ID">normed</span>=<span class="Number">1</span>)    <span class="Comment"># matplotlib version (plot)</span></span>
<span class="line"><span class="anchor" id="line-8-38"></span><span class="ID">pylab</span>.<span class="ID">show</span>()</span>
<span class="line"><span class="anchor" id="line-9-35"></span><span class="Comment"># Compute the histogram with numpy and then plot it</span></span>
<span class="line"><span class="anchor" id="line-10-32"></span>(<span class="ID">n</span>, <span class="ID">bins</span>) = <span class="ID">numpy</span>.<span class="ID">histogram</span>(<span class="ID">v</span>, <span class="ID">bins</span>=<span class="Number">50</span>, <span class="ID">normed</span>=<span class="ResWord">True</span>)<span class="Comment"># NumPy version (no plot)</span></span>
<span class="line"><span class="anchor" id="line-11-27"></span><span class="ID">pylab</span>.<span class="ID">plot</span>(.<span class="Number">5</span>*(<span class="ID">bins</span>[<span class="Number">1</span>:]+<span class="ID">bins</span>[:-<span class="Number">1</span>]), <span class="ID">n</span>)</span>
<span class="line"><span class="anchor" id="line-12-25"></span><span class="ID">pylab</span>.<span class="ID">show</span>()</span>
</pre></div></div><span class="anchor" id="line-1257"></span>
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