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<article class="bd-article">
<section id="glossary">
<h1>Glossary<a class="headerlink" href="#glossary" title="Link to this heading">#</a></h1>
<dl class="glossary">
<dt id="term-n">(<em class="xref py py-obj">n</em>,)<a class="headerlink" href="#term-n" title="Link to this term">#</a></dt><dd><p>A parenthesized number followed by a comma denotes a tuple with one
element. The trailing comma distinguishes a one-element tuple from a
parenthesized <code class="docutils literal notranslate"><span class="pre">n</span></code>.</p>
</dd>
<dt id="term-1">-1<a class="headerlink" href="#term-1" title="Link to this term">#</a></dt><dd><ul>
<li><p><strong>In a dimension entry</strong>, instructs NumPy to choose the length
that will keep the total number of array elements the same.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">12</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(4, 3)</span>
</pre></div>
</div>
</li>
<li><p><strong>In an index</strong>, any negative value
<a class="reference external" href="https://docs.python.org/dev/faq/programming.html#what-s-a-negative-index">denotes</a>
indexing from the right.</p></li>
</ul>
</dd>
<dt id="term-.-.-.">…<a class="headerlink" href="#term-.-.-." title="Link to this term">#</a></dt><dd><p>An <a class="reference external" href="https://docs.python.org/3/library/constants.html#Ellipsis" title="(in Python v3.14)"><code class="xref py py-data docutils literal notranslate"><span class="pre">Ellipsis</span></code></a>.</p>
<ul>
<li><p><strong>When indexing an array</strong>, shorthand that the missing axes, if they
exist, are full slices.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">24</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span><span class="p">[</span><span class="o">...</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(2, 3, 4)</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span><span class="p">[</span><span class="o">...</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(2, 3)</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="o">...</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(3, 4)</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="o">...</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(3,)</span>
</pre></div>
</div>
<p>It can be used at most once; <code class="docutils literal notranslate"><span class="pre">a[...,0,...]</span></code> raises an <a class="reference external" href="https://docs.python.org/3/library/exceptions.html#IndexError" title="(in Python v3.14)"><code class="xref py py-exc docutils literal notranslate"><span class="pre">IndexError</span></code></a>.</p>
</li>
<li><p><strong>In printouts</strong>, NumPy substitutes <code class="docutils literal notranslate"><span class="pre">...</span></code> for the middle elements of
large arrays. To see the entire array, use <a class="reference internal" href="reference/generated/numpy.printoptions.html#numpy.printoptions" title="numpy.printoptions"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.printoptions</span></code></a></p></li>
</ul>
</dd>
<dt id="term-0">:<a class="headerlink" href="#term-0" title="Link to this term">#</a></dt><dd><p>The Python <a class="reference external" href="https://docs.python.org/3/glossary.html#term-slice" title="(in Python v3.14)"><span>slice</span></a>
operator. In ndarrays, slicing can be applied to every
axis:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">24</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">array([[[ 0, 1, 2, 3],</span>
<span class="go"> [ 4, 5, 6, 7],</span>
<span class="go"> [ 8, 9, 10, 11]],</span>
<span class="go"> [[12, 13, 14, 15],</span>
<span class="go"> [16, 17, 18, 19],</span>
<span class="go"> [20, 21, 22, 23]]])</span>
<span class="gp">>>> </span><span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">:,</span><span class="o">-</span><span class="mi">2</span><span class="p">:,:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="go">array([[[16, 17, 18],</span>
<span class="go"> [20, 21, 22]]])</span>
</pre></div>
</div>
<p>Trailing slices can be omitted:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">,:,:]</span>
<span class="go">array([[ True, True, True, True],</span>
<span class="go"> [ True, True, True, True],</span>
<span class="go"> [ True, True, True, True]])</span>
</pre></div>
</div>
<p>In contrast to Python, where slicing creates a copy, in NumPy slicing
creates a <a class="reference internal" href="#term-view"><span class="xref std std-term">view</span></a>.</p>
<p>For details, see <a class="reference internal" href="user/basics.indexing.html#combining-advanced-and-basic-indexing"><span class="std std-ref">Combining advanced and basic indexing</span></a>.</p>
</dd>
<dt id="term-2"><<a class="headerlink" href="#term-2" title="Link to this term">#</a></dt><dd><p>In a dtype declaration, indicates that the data is
<a class="reference internal" href="#term-little-endian"><span class="xref std std-term">little-endian</span></a> (the bracket is big on the right).</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">dt</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dtype</span><span class="p">(</span><span class="s1">'<f'</span><span class="p">)</span> <span class="c1"># little-endian single-precision float</span>
</pre></div>
</div>
</dd>
<dt id="term-3">><a class="headerlink" href="#term-3" title="Link to this term">#</a></dt><dd><p>In a dtype declaration, indicates that the data is
<a class="reference internal" href="#term-big-endian"><span class="xref std std-term">big-endian</span></a> (the bracket is big on the left).</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">dt</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dtype</span><span class="p">(</span><span class="s1">'>H'</span><span class="p">)</span> <span class="c1"># big-endian unsigned short</span>
</pre></div>
</div>
</dd>
<dt id="term-advanced-indexing">advanced indexing<a class="headerlink" href="#term-advanced-indexing" title="Link to this term">#</a></dt><dd><p>Rather than using a <a class="reference internal" href="reference/arrays.scalars.html"><span class="doc">scalar</span></a> or slice as
an index, an axis can be indexed with an array, providing fine-grained
selection. This is known as <a class="reference internal" href="user/basics.indexing.html#advanced-indexing"><span class="std std-ref">advanced indexing</span></a>
or “fancy indexing”.</p>
</dd>
<dt id="term-along-an-axis">along an axis<a class="headerlink" href="#term-along-an-axis" title="Link to this term">#</a></dt><dd><p>An operation <em class="xref py py-obj">along axis n</em> of array <code class="docutils literal notranslate"><span class="pre">a</span></code> behaves as if its argument
were an array of slices of <code class="docutils literal notranslate"><span class="pre">a</span></code> where each slice has a successive
index of axis <em class="xref py py-obj">n</em>.</p>
<p>For example, if <code class="docutils literal notranslate"><span class="pre">a</span></code> is a 3 x <em class="xref py py-obj">N</em> array, an operation along axis 0
behaves as if its argument were an array containing slices of each row:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">((</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">,:],</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">,:],</span> <span class="n">a</span><span class="p">[</span><span class="mi">2</span><span class="p">,:]))</span>
</pre></div>
</div>
<p>To make it concrete, we can pick the operation to be the array-reversal
function <a class="reference internal" href="reference/generated/numpy.flip.html#numpy.flip" title="numpy.flip"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.flip</span></code></a>, which accepts an <code class="docutils literal notranslate"><span class="pre">axis</span></code> argument. We
construct a 3 x 4 array <code class="docutils literal notranslate"><span class="pre">a</span></code>:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">12</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">array([[ 0, 1, 2, 3],</span>
<span class="go"> [ 4, 5, 6, 7],</span>
<span class="go"> [ 8, 9, 10, 11]])</span>
</pre></div>
</div>
<p>Reversing along axis 0 (the row axis) yields</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">flip</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="go">array([[ 8, 9, 10, 11],</span>
<span class="go"> [ 4, 5, 6, 7],</span>
<span class="go"> [ 0, 1, 2, 3]])</span>
</pre></div>
</div>
<p>Recalling the definition of <em class="xref py py-obj">along an axis</em>, <code class="docutils literal notranslate"><span class="pre">flip</span></code> along axis 0 is
treating its argument as if it were</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">((</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">,:],</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">,:],</span> <span class="n">a</span><span class="p">[</span><span class="mi">2</span><span class="p">,:]))</span>
<span class="go">array([[ 0, 1, 2, 3],</span>
<span class="go"> [ 4, 5, 6, 7],</span>
<span class="go"> [ 8, 9, 10, 11]])</span>
</pre></div>
</div>
<p>and the result of <code class="docutils literal notranslate"><span class="pre">np.flip(a,axis=0)</span></code> is to reverse the slices:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">((</span><span class="n">a</span><span class="p">[</span><span class="mi">2</span><span class="p">,:],</span><span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">,:],</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]))</span>
<span class="go">array([[ 8, 9, 10, 11],</span>
<span class="go"> [ 4, 5, 6, 7],</span>
<span class="go"> [ 0, 1, 2, 3]])</span>
</pre></div>
</div>
</dd>
<dt id="term-array">array<a class="headerlink" href="#term-array" title="Link to this term">#</a></dt><dd><p>Used synonymously in the NumPy docs with <a class="reference internal" href="#term-ndarray"><span class="xref std std-term">ndarray</span></a>.</p>
</dd>
<dt id="term-array_like">array_like<a class="headerlink" href="#term-array_like" title="Link to this term">#</a></dt><dd><p>Any <a class="reference internal" href="reference/arrays.scalars.html"><span class="doc">scalar</span></a> or
<a class="reference external" href="https://docs.python.org/3/glossary.html#term-sequence" title="(in Python v3.14)"><span>sequence</span></a>
that can be interpreted as an ndarray. In addition to ndarrays
and scalars this category includes lists (possibly nested and with
different element types) and tuples. Any argument accepted by
<a class="reference internal" href="reference/generated/numpy.array.html"><span class="doc">numpy.array</span></a>
is array_like.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">1</span><span class="n">j</span><span class="p">,</span> <span class="mf">3.</span><span class="p">)])</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">array([[1.+0.j, 2.+0.j],</span>
<span class="go"> [0.+0.j, 0.+0.j],</span>
<span class="go"> [1.+1.j, 3.+0.j]])</span>
</pre></div>
</div>
</dd>
<dt id="term-array-scalar">array scalar<a class="headerlink" href="#term-array-scalar" title="Link to this term">#</a></dt><dd><p>An <a class="reference internal" href="reference/arrays.scalars.html"><span class="doc">array scalar</span></a> is an instance of the types/classes float32, float64,
etc.. For uniformity in handling operands, NumPy treats a scalar as
an array of zero dimension. In contrast, a 0-dimensional array is an <a class="reference internal" href="reference/arrays.ndarray.html"><span class="doc">ndarray</span></a> instance
containing precisely one value.</p>
</dd>
<dt id="term-axis">axis<a class="headerlink" href="#term-axis" title="Link to this term">#</a></dt><dd><p>Another term for an array dimension. Axes are numbered left to right;
axis 0 is the first element in the shape tuple.</p>
<p>In a two-dimensional vector, the elements of axis 0 are rows and the
elements of axis 1 are columns.</p>
<p>In higher dimensions, the picture changes. NumPy prints
higher-dimensional vectors as replications of row-by-column building
blocks, as in this three-dimensional vector:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">12</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">array([[[ 0, 1, 2],</span>
<span class="go"> [ 3, 4, 5]],</span>
<span class="go"> [[ 6, 7, 8],</span>
<span class="go"> [ 9, 10, 11]]])</span>
</pre></div>
</div>
<p><code class="docutils literal notranslate"><span class="pre">a</span></code> is depicted as a two-element array whose elements are 2x3 vectors.
From this point of view, rows and columns are the final two axes,
respectively, in any shape.</p>
<p>This rule helps you anticipate how a vector will be printed, and
conversely how to find the index of any of the printed elements. For
instance, in the example, the last two values of 8’s index must be 0 and
2. Since 8 appears in the second of the two 2x3’s, the first index must
be 1:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
<span class="go">8</span>
</pre></div>
</div>
<p>A convenient way to count dimensions in a printed vector is to
count <code class="docutils literal notranslate"><span class="pre">[</span></code> symbols after the open-parenthesis. This is
useful in distinguishing, say, a (1,2,3) shape from a (2,3) shape:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span><span class="o">.</span><span class="n">ndim</span>
<span class="go">2</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">array([[0, 1, 2],</span>
<span class="go"> [3, 4, 5]])</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span><span class="o">.</span><span class="n">ndim</span>
<span class="go">3</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">array([[[0, 1, 2],</span>
<span class="go"> [3, 4, 5]]])</span>
</pre></div>
</div>
</dd>
<dt id="term-.base">.base<a class="headerlink" href="#term-.base" title="Link to this term">#</a></dt><dd><p>If an array does not own its memory, then its
<a class="reference internal" href="reference/generated/numpy.ndarray.base.html"><span class="doc">base</span></a> attribute returns
the object whose memory the array is referencing. That object may be
referencing the memory from still another object, so the owning object
may be <code class="docutils literal notranslate"><span class="pre">a.base.base.base...</span></code>. Some writers erroneously claim that
testing <code class="docutils literal notranslate"><span class="pre">base</span></code> determines if arrays are <a class="reference internal" href="#term-view"><span class="xref std std-term">view</span></a>s. For the
correct way, see <a class="reference internal" href="reference/generated/numpy.shares_memory.html#numpy.shares_memory" title="numpy.shares_memory"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.shares_memory</span></code></a>.</p>
</dd>
<dt id="term-big-endian">big-endian<a class="headerlink" href="#term-big-endian" title="Link to this term">#</a></dt><dd><p>See <a class="reference external" href="https://en.wikipedia.org/wiki/Endianness">Endianness</a>.</p>
</dd>
<dt id="term-BLAS">BLAS<a class="headerlink" href="#term-BLAS" title="Link to this term">#</a></dt><dd><p><a class="reference external" href="https://en.wikipedia.org/wiki/Basic_Linear_Algebra_Subprograms">Basic Linear Algebra Subprograms</a></p>
</dd>
<dt id="term-broadcast">broadcast<a class="headerlink" href="#term-broadcast" title="Link to this term">#</a></dt><dd><p><em>broadcasting</em> is NumPy’s ability to process ndarrays of
different sizes as if all were the same size.</p>
<p>It permits an elegant do-what-I-mean behavior where, for instance,
adding a scalar to a vector adds the scalar value to every element.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">array([0, 1, 2])</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">+</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>
<span class="go">array([3, 4, 5])</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">+</span> <span class="mi">3</span>
<span class="go">array([3, 4, 5])</span>
</pre></div>
</div>
<p>Ordinarily, vector operands must all be the same size, because NumPy
works element by element – for instance, <code class="docutils literal notranslate"><span class="pre">c</span> <span class="pre">=</span> <span class="pre">a</span> <span class="pre">*</span> <span class="pre">b</span></code> is</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="n">c</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
<span class="n">c</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span>
<span class="o">...</span>
</pre></div>
</div>
<p>But in certain useful cases, NumPy can duplicate data along “missing”
axes or “too-short” dimensions so shapes will match. The duplication
costs no memory or time. For details, see
<a class="reference internal" href="user/basics.broadcasting.html"><span class="doc">Broadcasting.</span></a></p>
</dd>
<dt id="term-C-order">C order<a class="headerlink" href="#term-C-order" title="Link to this term">#</a></dt><dd><p>Same as <a class="reference internal" href="#term-row-major"><span class="xref std std-term">row-major</span></a>.</p>
</dd>
<dt id="term-casting">casting<a class="headerlink" href="#term-casting" title="Link to this term">#</a></dt><dd><p>The process of converting array data from one dtype to another. There
exist several casting modes, defined by the following casting rules:</p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">no</span></code>: The data types should not be cast at all.
Any mismatch in data types between the arrays will raise a
<em class="xref py py-obj">TypeError</em>.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">equiv</span></code>: Only byte-order changes are allowed.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">safe</span></code>: Only casts that can preserve values are allowed. Upcasting
(e.g., from int to float) is allowed, but downcasting is not.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">same_kind</span></code>: The ‘same_kind’ casting option allows safe casts and
casts within a kind, like float64 to float32.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">unsafe</span></code>: any data conversions may be done.</p></li>
</ul>
</dd>
<dt id="term-column-major">column-major<a class="headerlink" href="#term-column-major" title="Link to this term">#</a></dt><dd><p>See <a class="reference external" href="https://en.wikipedia.org/wiki/Row-_and_column-major_order">Row- and column-major order</a>.</p>
</dd>
<dt id="term-contiguous">contiguous<a class="headerlink" href="#term-contiguous" title="Link to this term">#</a></dt><dd><p>An array is contiguous if:</p>
<ul class="simple">
<li><p>it occupies an unbroken block of memory, and</p></li>
<li><p>array elements with higher indexes occupy higher addresses (that
is, no <a class="reference internal" href="#term-stride"><span class="xref std std-term">stride</span></a> is negative).</p></li>
</ul>
<p>There are two types of proper-contiguous NumPy arrays:</p>
<ul class="simple">
<li><p>Fortran-contiguous arrays refer to data that is stored column-wise,
i.e. the indexing of data as stored in memory starts from the
lowest dimension;</p></li>
<li><p>C-contiguous, or simply contiguous arrays, refer to data that is
stored row-wise, i.e. the indexing of data as stored in memory
starts from the highest dimension.</p></li>
</ul>
<p>For one-dimensional arrays these notions coincide.</p>
<p>For example, a 2x2 array <code class="docutils literal notranslate"><span class="pre">A</span></code> is Fortran-contiguous if its elements are
stored in memory in the following order:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">A</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="n">A</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="n">A</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span> <span class="n">A</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span>
</pre></div>
</div>
<p>and C-contiguous if the order is as follows:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">A</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="n">A</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span> <span class="n">A</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="n">A</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span>
</pre></div>
</div>
<p>To test whether an array is C-contiguous, use the <code class="docutils literal notranslate"><span class="pre">.flags.c_contiguous</span></code>
attribute of NumPy arrays. To test for Fortran contiguity, use the
<code class="docutils literal notranslate"><span class="pre">.flags.f_contiguous</span></code> attribute.</p>
</dd>
<dt id="term-copy">copy<a class="headerlink" href="#term-copy" title="Link to this term">#</a></dt><dd><p>See <a class="reference internal" href="#term-view"><span class="xref std std-term">view</span></a>.</p>
</dd>
<dt id="term-dimension">dimension<a class="headerlink" href="#term-dimension" title="Link to this term">#</a></dt><dd><p>See <a class="reference internal" href="#term-axis"><span class="xref std std-term">axis</span></a>.</p>
</dd>
<dt id="term-dtype">dtype<a class="headerlink" href="#term-dtype" title="Link to this term">#</a></dt><dd><p>The datatype describing the (identically typed) elements in an ndarray.
It can be changed to reinterpret the array contents. For details, see
<a class="reference internal" href="reference/arrays.dtypes.html"><span class="doc">Data type objects (dtype).</span></a></p>
</dd>
<dt id="term-fancy-indexing">fancy indexing<a class="headerlink" href="#term-fancy-indexing" title="Link to this term">#</a></dt><dd><p>Another term for <a class="reference internal" href="#term-advanced-indexing"><span class="xref std std-term">advanced indexing</span></a>.</p>
</dd>
<dt id="term-field">field<a class="headerlink" href="#term-field" title="Link to this term">#</a></dt><dd><p>In a <a class="reference internal" href="#term-structured-data-type"><span class="xref std std-term">structured data type</span></a>, each subtype is called a <em class="xref py py-obj">field</em>.
The <em class="xref py py-obj">field</em> has a name (a string), a type (any valid dtype), and
an optional <em class="xref py py-obj">title</em>. See <a class="reference internal" href="reference/arrays.dtypes.html#arrays-dtypes"><span class="std std-ref">Data type objects (dtype)</span></a>.</p>
</dd>
<dt id="term-Fortran-order">Fortran order<a class="headerlink" href="#term-Fortran-order" title="Link to this term">#</a></dt><dd><p>Same as <a class="reference internal" href="#term-column-major"><span class="xref std std-term">column-major</span></a>.</p>
</dd>
<dt id="term-flattened">flattened<a class="headerlink" href="#term-flattened" title="Link to this term">#</a></dt><dd><p>See <a class="reference internal" href="#term-ravel"><span class="xref std std-term">ravel</span></a>.</p>
</dd>
<dt id="term-homogeneous">homogeneous<a class="headerlink" href="#term-homogeneous" title="Link to this term">#</a></dt><dd><p>All elements of a homogeneous array have the same type. ndarrays, in
contrast to Python lists, are homogeneous. The type can be complicated,
as in a <a class="reference internal" href="#term-structured-array"><span class="xref std std-term">structured array</span></a>, but all elements have that type.</p>
<p>NumPy <a class="reference external" href="#term-object-array">object arrays</a>, which contain references to
Python objects, fill the role of heterogeneous arrays.</p>
</dd>
<dt id="term-itemsize">itemsize<a class="headerlink" href="#term-itemsize" title="Link to this term">#</a></dt><dd><p>The size of the dtype element in bytes.</p>
</dd>
<dt id="term-little-endian">little-endian<a class="headerlink" href="#term-little-endian" title="Link to this term">#</a></dt><dd><p>See <a class="reference external" href="https://en.wikipedia.org/wiki/Endianness">Endianness</a>.</p>
</dd>
<dt id="term-mask">mask<a class="headerlink" href="#term-mask" title="Link to this term">#</a></dt><dd><p>A boolean array used to select only certain elements for an operation:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">x</span>
<span class="go">array([0, 1, 2, 3, 4])</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">mask</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">></span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">mask</span>
<span class="go">array([False, False, False, True, True])</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
<span class="gp">>>> </span><span class="n">x</span>
<span class="go">array([ 0, 1, 2, -1, -1])</span>
</pre></div>
</div>
</dd>
<dt id="term-masked-array">masked array<a class="headerlink" href="#term-masked-array" title="Link to this term">#</a></dt><dd><p>Bad or missing data can be cleanly ignored by putting it in a masked
array, which has an internal boolean array indicating invalid
entries. Operations with masked arrays ignore these entries.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ma</span><span class="o">.</span><span class="n">masked_array</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">],</span> <span class="p">[</span><span class="kc">True</span><span class="p">,</span> <span class="kc">False</span><span class="p">,</span> <span class="kc">True</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">masked_array(data=[--, 2.0, --],</span>
<span class="go"> mask=[ True, False, True],</span>
<span class="go"> fill_value=1e+20)</span>
<span class="gp">>>> </span><span class="n">a</span> <span class="o">+</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>
<span class="go">masked_array(data=[--, 4.0, --],</span>
<span class="go"> mask=[ True, False, True],</span>
<span class="go"> fill_value=1e+20)</span>
</pre></div>
</div>
<p>For details, see <a class="reference internal" href="reference/maskedarray.html"><span class="doc">Masked arrays.</span></a></p>
</dd>
<dt id="term-matrix">matrix<a class="headerlink" href="#term-matrix" title="Link to this term">#</a></dt><dd><p>NumPy’s two-dimensional
<a class="reference internal" href="reference/generated/numpy.matrix.html"><span class="doc">matrix class</span></a>
should no longer be used; use regular ndarrays.</p>
</dd>
<dt id="term-ndarray">ndarray<a class="headerlink" href="#term-ndarray" title="Link to this term">#</a></dt><dd><p><a class="reference internal" href="reference/arrays.html"><span class="doc">NumPy’s basic structure</span></a>.</p>
</dd>
<dt id="term-object-array">object array<a class="headerlink" href="#term-object-array" title="Link to this term">#</a></dt><dd><p>An array whose dtype is <code class="docutils literal notranslate"><span class="pre">object</span></code>; that is, it contains references to
Python objects. Indexing the array dereferences the Python objects, so
unlike other ndarrays, an object array has the ability to hold
heterogeneous objects.</p>
</dd>
<dt id="term-ravel">ravel<a class="headerlink" href="#term-ravel" title="Link to this term">#</a></dt><dd><p><a class="reference internal" href="reference/generated/numpy.ravel.html"><span class="doc">numpy.ravel </span></a>
and <a class="reference internal" href="reference/generated/numpy.ndarray.flatten.html"><span class="doc">numpy.flatten </span></a>
both flatten an ndarray. <code class="docutils literal notranslate"><span class="pre">ravel</span></code> will return a view if possible;
<code class="docutils literal notranslate"><span class="pre">flatten</span></code> always returns a copy.</p>
<p>Flattening collapses a multidimensional array to a single dimension;
details of how this is done (for instance, whether <code class="docutils literal notranslate"><span class="pre">a[n+1]</span></code> should be
the next row or next column) are parameters.</p>
</dd>
<dt id="term-record-array">record array<a class="headerlink" href="#term-record-array" title="Link to this term">#</a></dt><dd><p>A <a class="reference internal" href="#term-structured-array"><span class="xref std std-term">structured array</span></a> with allowing access in an attribute style
(<code class="docutils literal notranslate"><span class="pre">a.field</span></code>) in addition to <code class="docutils literal notranslate"><span class="pre">a['field']</span></code>. For details, see
<a class="reference internal" href="reference/generated/numpy.recarray.html"><span class="doc">numpy.recarray.</span></a></p>
</dd>
<dt id="term-row-major">row-major<a class="headerlink" href="#term-row-major" title="Link to this term">#</a></dt><dd><p>See <a class="reference external" href="https://en.wikipedia.org/wiki/Row-_and_column-major_order">Row- and column-major order</a>.
NumPy creates arrays in row-major order by default.</p>
</dd>
<dt id="term-scalar">scalar<a class="headerlink" href="#term-scalar" title="Link to this term">#</a></dt><dd><p>In NumPy, usually a synonym for <a class="reference internal" href="#term-array-scalar"><span class="xref std std-term">array scalar</span></a>.</p>
</dd>
<dt id="term-shape">shape<a class="headerlink" href="#term-shape" title="Link to this term">#</a></dt><dd><p>A tuple showing the length of each dimension of an ndarray. The
length of the tuple itself is the number of dimensions
(<a class="reference internal" href="reference/generated/numpy.ndarray.ndim.html"><span class="doc">numpy.ndim</span></a>).
The product of the tuple elements is the number of elements in the
array. For details, see
<a class="reference internal" href="reference/generated/numpy.ndarray.shape.html"><span class="doc">numpy.ndarray.shape</span></a>.</p>
</dd>
<dt id="term-stride">stride<a class="headerlink" href="#term-stride" title="Link to this term">#</a></dt><dd><p>Physical memory is one-dimensional; strides provide a mechanism to map
a given index to an address in memory. For an N-dimensional array, its
<code class="docutils literal notranslate"><span class="pre">strides</span></code> attribute is an N-element tuple; advancing from index
<code class="docutils literal notranslate"><span class="pre">i</span></code> to index <code class="docutils literal notranslate"><span class="pre">i+1</span></code> on axis <code class="docutils literal notranslate"><span class="pre">n</span></code> means adding <code class="docutils literal notranslate"><span class="pre">a.strides[n]</span></code> bytes
to the address.</p>
<p>Strides are computed automatically from an array’s dtype and
shape, but can be directly specified using
<a class="reference internal" href="reference/generated/numpy.lib.stride_tricks.as_strided.html"><span class="doc">as_strided</span></a>.
Bounds validation can be enabled with the <code class="docutils literal notranslate"><span class="pre">check_bounds</span></code> parameter.</p>
<p>For details, see
<a class="reference internal" href="reference/generated/numpy.ndarray.strides.html"><span class="doc">numpy.ndarray.strides</span></a>.</p>
<p>To see how striding underlies the power of NumPy views, see
<a class="reference external" href="https://arxiv.org/pdf/1102.1523.pdf">The NumPy array: a structure for efficient numerical computation. </a></p>
</dd>
<dt id="term-structured-array">structured array<a class="headerlink" href="#term-structured-array" title="Link to this term">#</a></dt><dd><p>Array whose <a class="reference internal" href="#term-dtype"><span class="xref std std-term">dtype</span></a> is a <a class="reference internal" href="#term-structured-data-type"><span class="xref std std-term">structured data type</span></a>.</p>
</dd>
<dt id="term-structured-data-type">structured data type<a class="headerlink" href="#term-structured-data-type" title="Link to this term">#</a></dt><dd><p>Users can create arbitrarily complex <a class="reference internal" href="#term-dtype"><span class="xref std std-term">dtypes</span></a>
that can include other arrays and dtypes. These composite dtypes are called
<a class="reference internal" href="user/basics.rec.html"><span class="doc">structured data types.</span></a></p>
</dd>
<dt id="term-subarray">subarray<a class="headerlink" href="#term-subarray" title="Link to this term">#</a></dt><dd><p>An array nested in a <a class="reference internal" href="#term-structured-data-type"><span class="xref std std-term">structured data type</span></a>, as <code class="docutils literal notranslate"><span class="pre">b</span></code> is here:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">dt</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dtype</span><span class="p">([(</span><span class="s1">'a'</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">int32</span><span class="p">),</span> <span class="p">(</span><span class="s1">'b'</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">,</span> <span class="p">(</span><span class="mi">3</span><span class="p">,))])</span>
<span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dt</span><span class="p">)</span>
<span class="go">array([(0, [0., 0., 0.]), (0, [0., 0., 0.]), (0, [0., 0., 0.])],</span>
<span class="go"> dtype=[('a', '<i4'), ('b', '<f4', (3,))])</span>
</pre></div>
</div>
</dd>
<dt id="term-subarray-data-type">subarray data type<a class="headerlink" href="#term-subarray-data-type" title="Link to this term">#</a></dt><dd><p>An element of a structured datatype that behaves like an ndarray.</p>
</dd>
<dt id="term-title">title<a class="headerlink" href="#term-title" title="Link to this term">#</a></dt><dd><p>An alias for a field name in a structured datatype.</p>
</dd>
<dt id="term-type">type<a class="headerlink" href="#term-type" title="Link to this term">#</a></dt><dd><p>In NumPy, usually a synonym for <a class="reference internal" href="#term-dtype"><span class="xref std std-term">dtype</span></a>. For the more general
Python meaning, <a class="reference external" href="https://docs.python.org/3/glossary.html#term-type" title="(in Python v3.14)"><span class="xref std std-term">see here.</span></a></p>
</dd>
<dt id="term-ufunc">ufunc<a class="headerlink" href="#term-ufunc" title="Link to this term">#</a></dt><dd><p>NumPy’s fast element-by-element computation (<a class="reference internal" href="#term-vectorization"><span class="xref std std-term">vectorization</span></a>)
gives a choice which function gets applied. The general term for the
function is <code class="docutils literal notranslate"><span class="pre">ufunc</span></code>, short for <code class="docutils literal notranslate"><span class="pre">universal</span> <span class="pre">function</span></code>. NumPy routines
have built-in ufuncs, but users can also
<a class="reference internal" href="reference/ufuncs.html"><span class="doc">write their own.</span></a></p>
</dd>
<dt id="term-vectorization">vectorization<a class="headerlink" href="#term-vectorization" title="Link to this term">#</a></dt><dd><p>NumPy hands off array processing to C, where looping and computation are
much faster than in Python. To exploit this, programmers using NumPy
eliminate Python loops in favor of array-to-array operations.
<a class="reference internal" href="#term-vectorization"><span class="xref std std-term">vectorization</span></a> can refer both to the C offloading and to
structuring NumPy code to leverage it.</p>
</dd>
<dt id="term-view">view<a class="headerlink" href="#term-view" title="Link to this term">#</a></dt><dd><p>Without touching underlying data, NumPy can make one array appear
to change its datatype and shape.</p>
<p>An array created this way is a <em class="xref py py-obj">view</em>, and NumPy often exploits the
performance gain of using a view versus making a new array.</p>
<p>A potential drawback is that writing to a view can alter the original
as well. If this is a problem, NumPy instead needs to create a
physically distinct array – a <a class="reference external" href="https://docs.python.org/3/library/copy.html#module-copy" title="(in Python v3.14)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">copy</span></code></a>.</p>
<p>Some NumPy routines always return views, some always return copies, some
may return one or the other, and for some the choice can be specified.
Responsibility for managing views and copies falls to the programmer.
<a class="reference internal" href="reference/generated/numpy.shares_memory.html#numpy.shares_memory" title="numpy.shares_memory"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.shares_memory</span></code></a> will check whether <code class="docutils literal notranslate"><span class="pre">b</span></code> is a view of
<code class="docutils literal notranslate"><span class="pre">a</span></code>, but an exact answer isn’t always feasible, as the documentation
page explains.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">x</span>
<span class="go">array([0, 1, 2, 3, 4])</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="n">x</span><span class="p">[::</span><span class="mi">2</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">y</span>
<span class="go">array([0, 2, 4])</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">3</span> <span class="c1"># changing x changes y as well, since y is a view on x</span>
<span class="gp">>>> </span><span class="n">y</span>
<span class="go">array([3, 2, 4])</span>
</pre></div>
</div>
</dd>
</dl>
</section>
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