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<section id="nep-13-a-mechanism-for-overriding-ufuncs">
<span id="nep13"></span><h1>NEP 13 — A mechanism for overriding Ufuncs<a class="headerlink" href="#nep-13-a-mechanism-for-overriding-ufuncs" title="Link to this heading">#</a></h1>
<dl class="field-list simple">
<dt class="field-odd">Author<span class="colon">:</span></dt>
<dd class="field-odd"><p>Blake Griffith</p>
</dd>
<dt class="field-even">Contact<span class="colon">:</span></dt>
<dd class="field-even"><p><a class="reference external" href="mailto:blake.g%40utexas.edu">blake<span>.</span>g<span>@</span>utexas<span>.</span>edu</a></p>
</dd>
<dt class="field-odd">Date<span class="colon">:</span></dt>
<dd class="field-odd"><p>2013-07-10</p>
</dd>
<dt class="field-even">Author<span class="colon">:</span></dt>
<dd class="field-even"><p>Pauli Virtanen</p>
</dd>
<dt class="field-odd">Author<span class="colon">:</span></dt>
<dd class="field-odd"><p>Nathaniel Smith</p>
</dd>
<dt class="field-even">Author<span class="colon">:</span></dt>
<dd class="field-even"><p>Marten van Kerkwijk</p>
</dd>
<dt class="field-odd">Author<span class="colon">:</span></dt>
<dd class="field-odd"><p>Stephan Hoyer</p>
</dd>
<dt class="field-even">Date<span class="colon">:</span></dt>
<dd class="field-even"><p>2017-03-31</p>
</dd>
<dt class="field-odd">Status<span class="colon">:</span></dt>
<dd class="field-odd"><p>Final</p>
</dd>
<dt class="field-even">Updated<span class="colon">:</span></dt>
<dd class="field-even"><p>2023-02-19</p>
</dd>
<dt class="field-odd">Author<span class="colon">:</span></dt>
<dd class="field-odd"><p>Roy Smart</p>
</dd>
</dl>
<section id="executive-summary">
<h2>Executive summary<a class="headerlink" href="#executive-summary" title="Link to this heading">#</a></h2>
<p>NumPy’s universal functions (ufuncs) currently have some limited
functionality for operating on user defined subclasses of
<a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a> using <code class="docutils literal notranslate"><span class="pre">__array_prepare__</span></code> and <code class="docutils literal notranslate"><span class="pre">__array_wrap__</span></code>
<a class="footnote-reference brackets" href="#id5" id="id1" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a>, and there is little to no support for arbitrary
objects. e.g. SciPy’s sparse matrices <a class="footnote-reference brackets" href="#id6" id="id2" role="doc-noteref"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id7" id="id3" role="doc-noteref"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></a>.</p>
<p>Here we propose adding a mechanism to override ufuncs based on the ufunc
checking each of it’s arguments for a <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> method.
On discovery of <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> the ufunc will hand off the
operation to the method.</p>
<p>This covers some of the same ground as Travis Oliphant’s proposal to
retro-fit NumPy with multi-methods <a class="footnote-reference brackets" href="#id8" id="id4" role="doc-noteref"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></a>, which would solve the same
problem. The mechanism here follows more closely the way Python enables
classes to override <code class="docutils literal notranslate"><span class="pre">__mul__</span></code> and other binary operations. It also
specifically addresses how binary operators and ufuncs should interact.
(Note that in earlier iterations, the override was called
<code class="docutils literal notranslate"><span class="pre">__numpy_ufunc__</span></code>. An implementation was made, but had not quite the
right behaviour, hence the change in name.)</p>
<p>The <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> as described below requires that any
corresponding Python binary operations (<code class="docutils literal notranslate"><span class="pre">__mul__</span></code> et al.) should be
implemented in a specific way and be compatible with NumPy’s ndarray
semantics. Objects that do not satisfy this cannot override any NumPy
ufuncs. We do not specify a future-compatible path by which this
requirement can be relaxed — any changes here require corresponding
changes in 3rd party code.</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id5" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">1</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="http://docs.python.org/doc/numpy/user/basics.subclassing.html">http://docs.python.org/doc/numpy/user/basics.subclassing.html</a></p>
</aside>
<aside class="footnote brackets" id="id6" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id2">2</a><span class="fn-bracket">]</span></span>
<p><a class="github reference external" href="https://github.com/scipy/scipy/issues/2123">scipy/scipy#2123</a></p>
</aside>
<aside class="footnote brackets" id="id7" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">3</a><span class="fn-bracket">]</span></span>
<p><a class="github reference external" href="https://github.com/scipy/scipy/issues/1569">scipy/scipy#1569</a></p>
</aside>
<aside class="footnote brackets" id="id8" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id4">4</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://technicaldiscovery.blogspot.com/2013/07/thoughts-after-scipy-2013-and-specific.html">https://technicaldiscovery.blogspot.com/2013/07/thoughts-after-scipy-2013-and-specific.html</a></p>
</aside>
</aside>
</section>
<section id="motivation">
<h2>Motivation<a class="headerlink" href="#motivation" title="Link to this heading">#</a></h2>
<p>The current machinery for dispatching Ufuncs is generally agreed to be
insufficient. There have been lengthy discussions and other proposed
solutions <a class="footnote-reference brackets" href="#id11" id="id9" role="doc-noteref"><span class="fn-bracket">[</span>5<span class="fn-bracket">]</span></a>, <a class="footnote-reference brackets" href="#id12" id="id10" role="doc-noteref"><span class="fn-bracket">[</span>6<span class="fn-bracket">]</span></a>.</p>
<p>Using ufuncs with subclasses of <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a> is limited to
<code class="docutils literal notranslate"><span class="pre">__array_prepare__</span></code> and <code class="docutils literal notranslate"><span class="pre">__array_wrap__</span></code> to prepare the output arguments,
but these don’t allow you to for example change the shape or the data of
the arguments. Trying to ufunc things that don’t subclass
<a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a> is even more difficult, as the input arguments tend to
be cast to object arrays, which ends up producing surprising results.</p>
<p>Take this example of ufuncs interoperability with sparse matrices.:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">In</span> <span class="p">[</span><span class="mi">1</span><span class="p">]:</span> <span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="kn">import</span><span class="w"> </span><span class="nn">scipy.sparse</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">sp</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="n">size</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="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="n">size</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="n">asp</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">csr_matrix</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">bsp</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">csr_matrix</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="n">In</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="n">b</span>
<span class="n">Out</span><span class="p">[</span><span class="mi">2</span><span class="p">]:(</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span>
<span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
<span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">]]),</span>
<span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="p">[</span><span class="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="p">[</span><span class="mi">4</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">In</span> <span class="p">[</span><span class="mi">3</span><span class="p">]:</span> <span class="n">np</span><span class="o">.</span><span class="n">multiply</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="c1"># The right answer</span>
<span class="n">Out</span><span class="p">[</span><span class="mi">3</span><span class="p">]:</span> <span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
<span class="p">[</span><span class="mi">4</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">In</span> <span class="p">[</span><span class="mi">4</span><span class="p">]:</span> <span class="n">np</span><span class="o">.</span><span class="n">multiply</span><span class="p">(</span><span class="n">asp</span><span class="p">,</span> <span class="n">bsp</span><span class="p">)</span><span class="o">.</span><span class="n">todense</span><span class="p">()</span> <span class="c1"># calls __mul__ which does matrix multi</span>
<span class="n">Out</span><span class="p">[</span><span class="mi">4</span><span class="p">]:</span> <span class="n">matrix</span><span class="p">([[</span><span class="mi">16</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span>
<span class="p">[</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span>
<span class="p">[</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">]],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">int64</span><span class="p">)</span>
<span class="n">In</span> <span class="p">[</span><span class="mi">5</span><span class="p">]:</span> <span class="n">np</span><span class="o">.</span><span class="n">multiply</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">bsp</span><span class="p">)</span> <span class="c1"># Returns NotImplemented to user, bad!</span>
<span class="n">Out</span><span class="p">[</span><span class="mi">5</span><span class="p">]:</span> <span class="bp">NotImplemented</span>
</pre></div>
</div>
<p>Returning <a class="reference external" href="https://docs.python.org/dev/library/constants.html#NotImplemented" title="(in Python v3.14)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">NotImplemented</span></code></a> to user should not happen. Moreover:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">In</span> <span class="p">[</span><span class="mi">6</span><span class="p">]:</span> <span class="n">np</span><span class="o">.</span><span class="n">multiply</span><span class="p">(</span><span class="n">asp</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="n">Out</span><span class="p">[</span><span class="mi">6</span><span class="p">]:</span> <span class="n">array</span><span class="p">([[</span> <span class="o"><</span><span class="mi">3</span><span class="n">x3</span> <span class="n">sparse</span> <span class="n">matrix</span> <span class="n">of</span> <span class="nb">type</span> <span class="s1">'<class '</span><span class="n">numpy</span><span class="o">.</span><span class="n">int64</span><span class="s1">'>'</span>
<span class="k">with</span> <span class="mi">8</span> <span class="n">stored</span> <span class="n">elements</span> <span class="ow">in</span> <span class="n">Compressed</span> <span class="n">Sparse</span> <span class="n">Row</span> <span class="nb">format</span><span class="o">></span><span class="p">,</span>
<span class="o"><</span><span class="mi">3</span><span class="n">x3</span> <span class="n">sparse</span> <span class="n">matrix</span> <span class="n">of</span> <span class="nb">type</span> <span class="s1">'<class '</span><span class="n">numpy</span><span class="o">.</span><span class="n">int64</span><span class="s1">'>'</span>
<span class="k">with</span> <span class="mi">8</span> <span class="n">stored</span> <span class="n">elements</span> <span class="ow">in</span> <span class="n">Compressed</span> <span class="n">Sparse</span> <span class="n">Row</span> <span class="nb">format</span><span class="o">></span><span class="p">,</span>
<span class="o"><</span><span class="mi">3</span><span class="n">x3</span> <span class="n">sparse</span> <span class="n">matrix</span> <span class="n">of</span> <span class="nb">type</span> <span class="s1">'<class '</span><span class="n">numpy</span><span class="o">.</span><span class="n">int64</span><span class="s1">'>'</span>
<span class="k">with</span> <span class="mi">8</span> <span class="n">stored</span> <span class="n">elements</span> <span class="ow">in</span> <span class="n">Compressed</span> <span class="n">Sparse</span> <span class="n">Row</span> <span class="nb">format</span><span class="o">></span><span class="p">],</span>
<span class="p">[</span> <span class="o"><</span><span class="mi">3</span><span class="n">x3</span> <span class="n">sparse</span> <span class="n">matrix</span> <span class="n">of</span> <span class="nb">type</span> <span class="s1">'<class '</span><span class="n">numpy</span><span class="o">.</span><span class="n">int64</span><span class="s1">'>'</span>
<span class="k">with</span> <span class="mi">8</span> <span class="n">stored</span> <span class="n">elements</span> <span class="ow">in</span> <span class="n">Compressed</span> <span class="n">Sparse</span> <span class="n">Row</span> <span class="nb">format</span><span class="o">></span><span class="p">,</span>
<span class="o"><</span><span class="mi">3</span><span class="n">x3</span> <span class="n">sparse</span> <span class="n">matrix</span> <span class="n">of</span> <span class="nb">type</span> <span class="s1">'<class '</span><span class="n">numpy</span><span class="o">.</span><span class="n">int64</span><span class="s1">'>'</span>
<span class="k">with</span> <span class="mi">8</span> <span class="n">stored</span> <span class="n">elements</span> <span class="ow">in</span> <span class="n">Compressed</span> <span class="n">Sparse</span> <span class="n">Row</span> <span class="nb">format</span><span class="o">></span><span class="p">,</span>
<span class="o"><</span><span class="mi">3</span><span class="n">x3</span> <span class="n">sparse</span> <span class="n">matrix</span> <span class="n">of</span> <span class="nb">type</span> <span class="s1">'<class '</span><span class="n">numpy</span><span class="o">.</span><span class="n">int64</span><span class="s1">'>'</span>
<span class="k">with</span> <span class="mi">8</span> <span class="n">stored</span> <span class="n">elements</span> <span class="ow">in</span> <span class="n">Compressed</span> <span class="n">Sparse</span> <span class="n">Row</span> <span class="nb">format</span><span class="o">></span><span class="p">],</span>
<span class="p">[</span> <span class="o"><</span><span class="mi">3</span><span class="n">x3</span> <span class="n">sparse</span> <span class="n">matrix</span> <span class="n">of</span> <span class="nb">type</span> <span class="s1">'<class '</span><span class="n">numpy</span><span class="o">.</span><span class="n">int64</span><span class="s1">'>'</span>
<span class="k">with</span> <span class="mi">8</span> <span class="n">stored</span> <span class="n">elements</span> <span class="ow">in</span> <span class="n">Compressed</span> <span class="n">Sparse</span> <span class="n">Row</span> <span class="nb">format</span><span class="o">></span><span class="p">,</span>
<span class="o"><</span><span class="mi">3</span><span class="n">x3</span> <span class="n">sparse</span> <span class="n">matrix</span> <span class="n">of</span> <span class="nb">type</span> <span class="s1">'<class '</span><span class="n">numpy</span><span class="o">.</span><span class="n">int64</span><span class="s1">'>'</span>
<span class="k">with</span> <span class="mi">8</span> <span class="n">stored</span> <span class="n">elements</span> <span class="ow">in</span> <span class="n">Compressed</span> <span class="n">Sparse</span> <span class="n">Row</span> <span class="nb">format</span><span class="o">></span><span class="p">,</span>
<span class="o"><</span><span class="mi">3</span><span class="n">x3</span> <span class="n">sparse</span> <span class="n">matrix</span> <span class="n">of</span> <span class="nb">type</span> <span class="s1">'<class '</span><span class="n">numpy</span><span class="o">.</span><span class="n">int64</span><span class="s1">'>'</span>
<span class="k">with</span> <span class="mi">8</span> <span class="n">stored</span> <span class="n">elements</span> <span class="ow">in</span> <span class="n">Compressed</span> <span class="n">Sparse</span> <span class="n">Row</span> <span class="nb">format</span><span class="o">></span><span class="p">]],</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">object</span><span class="p">)</span>
</pre></div>
</div>
<p>Here, it appears that the sparse matrix was converted to an object array
scalar, which was then multiplied with all elements of the <code class="docutils literal notranslate"><span class="pre">b</span></code> array.
However, this behavior is more confusing than useful, and having a
<a class="reference external" href="https://docs.python.org/dev/library/exceptions.html#TypeError" title="(in Python v3.14)"><code class="xref py py-exc docutils literal notranslate"><span class="pre">TypeError</span></code></a> would be preferable.</p>
<p>This proposal will <em>not</em> resolve the issue with scipy.sparse matrices,
which have multiplication semantics incompatible with NumPy arrays.
However, the aim is to enable writing other custom array types that have
strictly ndarray compatible semantics.</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id11" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id9">5</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://mail.python.org/pipermail/numpy-discussion/2011-June/056945.html">https://mail.python.org/pipermail/numpy-discussion/2011-June/056945.html</a></p>
</aside>
<aside class="footnote brackets" id="id12" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>6<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id10">1</a>,<a role="doc-backlink" href="#id18">2</a>)</span>
<p><a class="github reference external" href="https://github.com/numpy/numpy/issues/5844">numpy/numpy#5844</a></p>
</aside>
</aside>
</section>
<section id="proposed-interface">
<h2>Proposed interface<a class="headerlink" href="#proposed-interface" title="Link to this heading">#</a></h2>
<p>The standard array class <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a> gains an <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code>
method and objects can override Ufuncs by overriding this method (if
they are <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a> subclasses) or defining their own. The method
signature is:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">def</span><span class="w"> </span><span class="nf">__array_ufunc__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ufunc</span><span class="p">,</span> <span class="n">method</span><span class="p">,</span> <span class="o">*</span><span class="n">inputs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
</pre></div>
</div>
<p>Here:</p>
<ul class="simple">
<li><p><em>ufunc</em> is the ufunc object that was called.</p></li>
<li><p><em>method</em> is a string indicating how the Ufunc was called, either
<code class="docutils literal notranslate"><span class="pre">"__call__"</span></code> to indicate it was called directly, or one of its
methods: <code class="docutils literal notranslate"><span class="pre">"reduce"</span></code>, <code class="docutils literal notranslate"><span class="pre">"accumulate"</span></code>, <code class="docutils literal notranslate"><span class="pre">"reduceat"</span></code>, <code class="docutils literal notranslate"><span class="pre">"outer"</span></code>,
or <code class="docutils literal notranslate"><span class="pre">"at"</span></code>.</p></li>
<li><p><em>inputs</em> is a tuple of the input arguments to the <code class="docutils literal notranslate"><span class="pre">ufunc</span></code></p></li>
<li><p><em>kwargs</em> contains any optional or keyword arguments passed to the
function. This includes any <code class="docutils literal notranslate"><span class="pre">out</span></code> arguments, which are always
contained in a tuple.</p></li>
</ul>
<p>Hence, the arguments are normalized: only the required input arguments
(<code class="docutils literal notranslate"><span class="pre">inputs</span></code>) are passed on as positional arguments, all the others are
passed on as a dict of keyword arguments (<code class="docutils literal notranslate"><span class="pre">kwargs</span></code>). In particular, if
there are output arguments, positional are otherwise, that are not
<a class="reference external" href="https://docs.python.org/dev/library/constants.html#None" title="(in Python v3.14)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">None</span></code></a>, they are passed on as a tuple in the <code class="docutils literal notranslate"><span class="pre">out</span></code> keyword
argument (even for the <code class="docutils literal notranslate"><span class="pre">reduce</span></code>, <code class="docutils literal notranslate"><span class="pre">accumulate</span></code>, and <code class="docutils literal notranslate"><span class="pre">reduceat</span></code> methods
where in all current cases only a single output makes sense).</p>
<p>The function dispatch proceeds as follows:</p>
<ul class="simple">
<li><p>If one of the input, output, or <code class="docutils literal notranslate"><span class="pre">where</span></code> arguments implements
<code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code>, it is executed instead of the ufunc.</p></li>
<li><p>If more than one of the arguments implements <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code>,
they are tried in the following order: subclasses before superclasses,
inputs before outputs, outputs before <code class="docutils literal notranslate"><span class="pre">where</span></code>, otherwise left to right.</p></li>
<li><p>The first <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> method returning something else than
<a class="reference external" href="https://docs.python.org/dev/library/constants.html#NotImplemented" title="(in Python v3.14)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">NotImplemented</span></code></a> determines the return value of the Ufunc.</p></li>
<li><p>If all <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> methods of the input arguments return
<a class="reference external" href="https://docs.python.org/dev/library/constants.html#NotImplemented" title="(in Python v3.14)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">NotImplemented</span></code></a>, a <a class="reference external" href="https://docs.python.org/dev/library/exceptions.html#TypeError" title="(in Python v3.14)"><code class="xref py py-exc docutils literal notranslate"><span class="pre">TypeError</span></code></a> is raised.</p></li>
<li><p>If a <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> method raises an error, the error is
propagated immediately.</p></li>
<li><p>If none of the input arguments had an <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> method, the
execution falls back on the default ufunc behaviour.</p></li>
</ul>
<p>In the above, there is one proviso: if a class has an
<code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> attribute but it is identical to
<code class="docutils literal notranslate"><span class="pre">ndarray.__array_ufunc__</span></code>, the attribute is ignored. This happens for
instances of <cite>ndarray</cite> and for <cite>ndarray</cite> subclasses that did not
override their inherited <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> implementation.</p>
<section id="type-casting-hierarchy">
<h3>Type casting hierarchy<a class="headerlink" href="#type-casting-hierarchy" title="Link to this heading">#</a></h3>
<p>The Python operator override mechanism gives much freedom in how to
write the override methods, and it requires some discipline in order to
achieve predictable results. Here, we discuss an approach for
understanding some of the implications, which can provide input in the
design.</p>
<p>It is useful to maintain a clear idea of what types can be “upcast” to
others, possibly indirectly (e.g. indirect A->B->C is implemented but
direct A->C not). If the implementations of <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> follow a
coherent type casting hierarchy, it can be used to understand results of
operations.</p>
<p>Type casting can be expressed as a <a class="reference external" href="https://en.wikipedia.org/wiki/Graph_theory">graph</a>
defined as follows:</p>
<blockquote>
<div><p>For each <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> method, draw directed edges from each
possible input type to each possible output type.</p>
<p>That is, in each case where <code class="docutils literal notranslate"><span class="pre">y</span> <span class="pre">=</span> <span class="pre">x.__array_ufunc__(a,</span> <span class="pre">b,</span> <span class="pre">c,</span> <span class="pre">...)</span></code>
does something else than returning <code class="docutils literal notranslate"><span class="pre">NotImplemented</span></code> or raising an error,
draw edges <code class="docutils literal notranslate"><span class="pre">type(a)</span> <span class="pre">-></span> <span class="pre">type(y)</span></code>, <code class="docutils literal notranslate"><span class="pre">type(b)</span> <span class="pre">-></span> <span class="pre">type(y)</span></code>, …</p>
</div></blockquote>
<p>If the resulting graph is <em>acyclic</em>, it defines a coherent type casting
hierarchy (unambiguous partial ordering between types). In this case,
operations involving multiple types generally predictably produce result
of the “highest” type, or raise a <a class="reference external" href="https://docs.python.org/dev/library/exceptions.html#TypeError" title="(in Python v3.14)"><code class="xref py py-exc docutils literal notranslate"><span class="pre">TypeError</span></code></a>. See examples at the
end of this section.</p>
<p>If the graph has cycles, the <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> type casting is not
well-defined, and things such as <code class="docutils literal notranslate"><span class="pre">type(multiply(a,</span> <span class="pre">b))</span> <span class="pre">!=</span>
<span class="pre">type(multiply(b,</span> <span class="pre">a))</span></code> or <code class="docutils literal notranslate"><span class="pre">type(add(a,</span> <span class="pre">add(b,</span> <span class="pre">c)))</span> <span class="pre">!=</span> <span class="pre">type(add(add(a,</span>
<span class="pre">b),</span> <span class="pre">c))</span></code> are not excluded (and then probably always possible).</p>
<p>If the type casting hierarchy is well defined, for each class A, all
other classes that define <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> belong to exactly one of
three groups:</p>
<ul class="simple">
<li><p><em>Above A</em>: the types that A can be (indirectly) upcast to in ufuncs.</p></li>
<li><p><em>Below A</em>: the types that can be (indirectly) upcast to A in ufuncs.</p></li>
<li><p><em>Incompatible</em>: neither above nor below A; types for which no
(indirect) upcasting is possible.</p></li>
</ul>
<p>Note that the legacy behaviour of NumPy ufuncs is to try to convert
unknown objects to <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a> via <code class="xref py py-func docutils literal notranslate"><span class="pre">np.asarray()</span></code>. This is
equivalent to placing <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a> above these objects in the graph.
Since we above defined <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a> to return <cite>NotImplemented</cite> for
classes with custom <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code>, this puts <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a>
below such classes in the type hierarchy, allowing the operations to be
overridden.</p>
<p>In view of the above, binary ufuncs describing transitive operations
should aim to define a well-defined casting hierarchy. This is likely
also a sensible approach to all ufuncs — exceptions to this should
consider carefully if any surprising behavior results.</p>
<div class="admonition-example admonition">
<p class="admonition-title">Example</p>
<p>Type casting hierarchy.</p>
<img alt="_images/nep0013_image1.png" src="_images/nep0013_image1.png" />
<p>The <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> of type A can handle ndarrays returning C,
B can handle ndarray and D returning B, and C can handle A and B returning C,
but not ndarrays or D. The
result is a directed acyclic graph, and defines a type casting
hierarchy, with relations <code class="docutils literal notranslate"><span class="pre">C</span> <span class="pre">></span> <span class="pre">A</span></code>, <code class="docutils literal notranslate"><span class="pre">C</span> <span class="pre">></span> <span class="pre">ndarray</span></code>, <code class="docutils literal notranslate"><span class="pre">C</span> <span class="pre">></span> <span class="pre">B</span> <span class="pre">></span> <span class="pre">ndarray</span></code>,
<code class="docutils literal notranslate"><span class="pre">C</span> <span class="pre">></span> <span class="pre">B</span> <span class="pre">></span> <span class="pre">D</span></code>. The type A is incompatible with B, D, ndarray,
and D is incompatible with A and ndarray. Ufunc
expressions involving these classes should produce results of the
highest type involved or raise a <a class="reference external" href="https://docs.python.org/dev/library/exceptions.html#TypeError" title="(in Python v3.14)"><code class="xref py py-exc docutils literal notranslate"><span class="pre">TypeError</span></code></a>.</p>
</div>
<div class="admonition-example admonition">
<p class="admonition-title">Example</p>
<p>One-cycle in the <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> graph.</p>
<img alt="_images/nep0013_image2.png" src="_images/nep0013_image2.png" />
<p>In this case, the <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> relations have a cycle of length 1,
and a type casting hierarchy does not exist. Binary operations are not
commutative: <code class="docutils literal notranslate"><span class="pre">type(a</span> <span class="pre">+</span> <span class="pre">b)</span> <span class="pre">is</span> <span class="pre">A</span></code> but <code class="docutils literal notranslate"><span class="pre">type(b</span> <span class="pre">+</span> <span class="pre">a)</span> <span class="pre">is</span> <span class="pre">B</span></code>.</p>
</div>
<div class="admonition-example admonition">
<p class="admonition-title">Example</p>
<p>Longer cycle in the <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> graph.</p>
<img alt="_images/nep0013_image3.png" src="_images/nep0013_image3.png" />
<p>In this case, the <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> relations have a longer cycle, and a
type casting hierarchy does not exist. Binary operations are still
commutative, but type transitivity is lost: <code class="docutils literal notranslate"><span class="pre">type(a</span> <span class="pre">+</span> <span class="pre">(b</span> <span class="pre">+</span> <span class="pre">c))</span> <span class="pre">is</span> <span class="pre">A</span></code> but
<code class="docutils literal notranslate"><span class="pre">type((a</span> <span class="pre">+</span> <span class="pre">b)</span> <span class="pre">+</span> <span class="pre">c)</span> <span class="pre">is</span> <span class="pre">C</span></code>.</p>
</div>
</section>
<section id="subclass-hierarchies">
<h3>Subclass hierarchies<a class="headerlink" href="#subclass-hierarchies" title="Link to this heading">#</a></h3>
<p>Generally, it is desirable to mirror the class hierarchy in the ufunc
type casting hierarchy. The recommendation is that an
<code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> implementation of a class should generally return
<cite>NotImplemented</cite> unless the inputs are instances of the same class or
superclasses. This guarantees that in the type casting hierarchy,
superclasses are below, subclasses above, and other classes are
incompatible. Exceptions to this need to check they respect the
implicit type casting hierarchy.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Note that type casting hierarchy and class hierarchy are here defined
to go the “opposite” directions. It would in principle also be
consistent to have <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> handle also instances of
subclasses. In this case, the “subclasses first” dispatch rule would
ensure a relatively similar outcome. However, the behavior is then less
explicitly specified.</p>
</div>
<p>Subclasses can be easily constructed if methods consistently use
<code class="xref py py-func docutils literal notranslate"><span class="pre">super()</span></code> to pass through the class hierarchy <a class="footnote-reference brackets" href="#id14" id="id13" role="doc-noteref"><span class="fn-bracket">[</span>7<span class="fn-bracket">]</span></a>. To support
this, <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a> has its own <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> method,
equivalent to:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">def</span><span class="w"> </span><span class="nf">__array_ufunc__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ufunc</span><span class="p">,</span> <span class="n">method</span><span class="p">,</span> <span class="o">*</span><span class="n">inputs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
<span class="c1"># Cannot handle items that have __array_ufunc__ (other than our own).</span>
<span class="n">outputs</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">'out'</span><span class="p">,</span> <span class="p">())</span>
<span class="n">objs</span> <span class="o">=</span> <span class="n">inputs</span> <span class="o">+</span> <span class="n">outputs</span>
<span class="k">if</span> <span class="s2">"where"</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
<span class="n">objs</span> <span class="o">=</span> <span class="n">objs</span> <span class="o">+</span> <span class="p">(</span><span class="n">kwargs</span><span class="p">[</span><span class="s2">"where"</span><span class="p">],</span> <span class="p">)</span>
<span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">objs</span><span class="p">:</span>
<span class="k">if</span> <span class="p">(</span><span class="nb">hasattr</span><span class="p">(</span><span class="n">item</span><span class="p">,</span> <span class="s1">'__array_ufunc__'</span><span class="p">)</span> <span class="ow">and</span>
<span class="nb">type</span><span class="p">(</span><span class="n">item</span><span class="p">)</span><span class="o">.</span><span class="n">__array_ufunc__</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">ndarray</span><span class="o">.</span><span class="n">__array_ufunc__</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">NotImplemented</span>
<span class="c1"># If we didn't have to support legacy behaviour (__array_prepare__,</span>
<span class="c1"># __array_wrap__, etc.), we might here convert python floats,</span>
<span class="c1"># lists, etc, to arrays with</span>
<span class="c1"># items = [np.asarray(item) for item in inputs]</span>
<span class="c1"># and then start the right iterator for the given method.</span>
<span class="c1"># However, we do have to support legacy, so call back into the ufunc.</span>
<span class="c1"># Its arguments are now guaranteed not to have __array_ufunc__</span>
<span class="c1"># overrides, and it will do the coercion to array for us.</span>
<span class="k">return</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">ufunc</span><span class="p">,</span> <span class="n">method</span><span class="p">)(</span><span class="o">*</span><span class="n">items</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
</pre></div>
</div>
<p>Note that, as a special case, the ufunc dispatch mechanism does not call
this <cite>ndarray.__array_ufunc__</cite> method, even for <cite>ndarray</cite> subclasses
if they have not overridden the default <cite>ndarray</cite> implementation. As a
consequence, calling <cite>ndarray.__array_ufunc__</cite> will not result to a
nested ufunc dispatch cycle.</p>
<p>The use of <code class="xref py py-func docutils literal notranslate"><span class="pre">super()</span></code> should be particularly useful for subclasses of
<a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a> that only add an attribute like a unit. In their
<cite>__array_ufunc__</cite> implementation, such classes can do possible
adjustment of the arguments relevant to their own class, and pass on to
the superclass implementation using <code class="xref py py-func docutils literal notranslate"><span class="pre">super()</span></code> until the ufunc is
actually done, and then do possible adjustments of the outputs.</p>
<p>In general, custom implementations of <cite>__array_ufunc__</cite> should avoid
nested dispatch cycles, where one not just calls the ufunc via
<code class="docutils literal notranslate"><span class="pre">getattr(ufunc,</span> <span class="pre">method)(*items,</span> <span class="pre">**kwargs)</span></code>, but catches possible
exceptions, etc. As always, there may be exceptions. For instance, for a
class like <code class="xref py py-class docutils literal notranslate"><span class="pre">MaskedArray</span></code>, which only cares that whatever
it contains is an <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a> subclass, a reimplementation with
<code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> may well be more easily done by directly applying
the ufunc to its data, and then adjusting the mask. Indeed, one can
think of this as part of the class determining whether it can handle the
other argument (i.e., where in the type hierarchy it sits). In this
case, one should return <a class="reference external" href="https://docs.python.org/dev/library/constants.html#NotImplemented" title="(in Python v3.14)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">NotImplemented</span></code></a> if the trial fails. So,
the implementation would be something like:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">def</span><span class="w"> </span><span class="nf">__array_ufunc__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ufunc</span><span class="p">,</span> <span class="n">method</span><span class="p">,</span> <span class="o">*</span><span class="n">inputs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
<span class="c1"># for simplicity, outputs are ignored here.</span>
<span class="n">unmasked_items</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">((</span><span class="n">item</span><span class="o">.</span><span class="n">data</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">item</span><span class="p">,</span> <span class="n">MaskedArray</span><span class="p">)</span>
<span class="k">else</span> <span class="n">item</span><span class="p">)</span> <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">inputs</span><span class="p">)</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">unmasked_result</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">ufunc</span><span class="p">,</span> <span class="n">method</span><span class="p">)(</span><span class="o">*</span><span class="n">unmasked_items</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
<span class="k">return</span> <span class="bp">NotImplemented</span>
<span class="c1"># for simplicity, ignore that unmasked_result could be a tuple</span>
<span class="c1"># or a scalar.</span>
<span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">unmasked_result</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">NotImplemented</span>
<span class="c1"># now combine masks and view as MaskedArray instance</span>
<span class="o">...</span>
</pre></div>
</div>
<p>As a specific example, consider a quantity and a masked array class
which both override <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code>, with specific instances <code class="docutils literal notranslate"><span class="pre">q</span></code>
and <code class="docutils literal notranslate"><span class="pre">ma</span></code>, where the latter contains a regular array. Executing
<code class="docutils literal notranslate"><span class="pre">np.multiply(q,</span> <span class="pre">ma)</span></code>, the ufunc will first dispatch to
<code class="docutils literal notranslate"><span class="pre">q.__array_ufunc__</span></code>, which returns <a class="reference external" href="https://docs.python.org/dev/library/constants.html#NotImplemented" title="(in Python v3.14)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">NotImplemented</span></code></a> (since the
quantity class turns itself into an array and calls <code class="xref py py-func docutils literal notranslate"><span class="pre">super()</span></code>, which
passes on to <code class="docutils literal notranslate"><span class="pre">ndarray.__array_ufunc__</span></code>, which sees the override on
<code class="docutils literal notranslate"><span class="pre">ma</span></code>). Next, <code class="docutils literal notranslate"><span class="pre">ma.__array_ufunc__</span></code> gets a chance. It does not know
quantity, and if it were to just return <a class="reference external" href="https://docs.python.org/dev/library/constants.html#NotImplemented" title="(in Python v3.14)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">NotImplemented</span></code></a> as well,
an <a class="reference external" href="https://docs.python.org/dev/library/exceptions.html#TypeError" title="(in Python v3.14)"><code class="xref py py-exc docutils literal notranslate"><span class="pre">TypeError</span></code></a> would result. But in our sample implementation, it
uses <code class="docutils literal notranslate"><span class="pre">getattr(ufunc,</span> <span class="pre">method)</span></code> to, effectively, evaluate
<code class="docutils literal notranslate"><span class="pre">np.multiply(q,</span> <span class="pre">ma.data)</span></code>. This again will pass to
<code class="docutils literal notranslate"><span class="pre">q.__array_ufunc__</span></code>, but this time, since <code class="docutils literal notranslate"><span class="pre">ma.data</span></code> is a regular
array, it will return a result that is also a quantity. Since this is a
subclass of <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a>, <code class="docutils literal notranslate"><span class="pre">ma.__array_ufunc__</span></code> can turn this into
a masked array and thus return a result (obviously, if it was not a
array subclass, it could still return <a class="reference external" href="https://docs.python.org/dev/library/constants.html#NotImplemented" title="(in Python v3.14)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">NotImplemented</span></code></a>).</p>
<p>Note that in the context of the type hierarchy discussed above this is a
somewhat tricky example, since <code class="xref py py-class docutils literal notranslate"><span class="pre">MaskedArray</span></code> has a strange
position: it is above all subclasses of <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a>, in that it can
cast them to its own type, but it does not itself know how to interact
with them in ufuncs.</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id14" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id13">7</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://rhettinger.wordpress.com/2011/05/26/super-considered-super/">https://rhettinger.wordpress.com/2011/05/26/super-considered-super/</a></p>
</aside>
</aside>
</section>
<section id="turning-ufuncs-off">
<span id="neps-ufunc-overrides-turning-ufuncs-off"></span><h3>Turning Ufuncs off<a class="headerlink" href="#turning-ufuncs-off" title="Link to this heading">#</a></h3>
<p>For some classes, Ufuncs make no sense, and, like for some other special
methods such as <code class="docutils literal notranslate"><span class="pre">__hash__</span></code> and <code class="docutils literal notranslate"><span class="pre">__iter__</span></code> <a class="footnote-reference brackets" href="#id16" id="id15" role="doc-noteref"><span class="fn-bracket">[</span>8<span class="fn-bracket">]</span></a>, one can indicate
Ufuncs are not available by setting <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> to <a class="reference external" href="https://docs.python.org/dev/library/constants.html#None" title="(in Python v3.14)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">None</span></code></a>.
If a Ufunc is called on any operand that sets <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span> <span class="pre">=</span> <span class="pre">None</span></code>,
it will unconditionally raise <a class="reference external" href="https://docs.python.org/dev/library/exceptions.html#TypeError" title="(in Python v3.14)"><code class="xref py py-exc docutils literal notranslate"><span class="pre">TypeError</span></code></a>.</p>
<p>In the type casting hierarchy, this makes it explicit that the type is
incompatible relative to <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a>.</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id16" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id15">8</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://docs.python.org/3/reference/datamodel.html#specialnames">https://docs.python.org/3/reference/datamodel.html#specialnames</a></p>
</aside>
</aside>
</section>
<section id="behavior-in-combination-with-python-s-binary-operations">
<h3>Behavior in combination with Python’s binary operations<a class="headerlink" href="#behavior-in-combination-with-python-s-binary-operations" title="Link to this heading">#</a></h3>
<p>The Python operator override mechanism in <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a> is coupled to
the <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> mechanism. For the special methods calls such as
<code class="docutils literal notranslate"><span class="pre">ndarray.__mul__(self,</span> <span class="pre">other)</span></code> that Python calls for implementing
binary operations such as <code class="docutils literal notranslate"><span class="pre">*</span></code> and <code class="docutils literal notranslate"><span class="pre">+</span></code>, NumPy’s <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a>
implements the following behavior:</p>
<ul class="simple">
<li><p>If <code class="docutils literal notranslate"><span class="pre">other.__array_ufunc__</span> <span class="pre">is</span> <span class="pre">None</span></code>, <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a>
returns <a class="reference external" href="https://docs.python.org/dev/library/constants.html#NotImplemented" title="(in Python v3.14)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">NotImplemented</span></code></a>. Control reverts to Python, which in turn
will try calling a corresponding reflexive method on <code class="docutils literal notranslate"><span class="pre">other</span></code> (e.g.,
<code class="docutils literal notranslate"><span class="pre">other.__rmul__</span></code>), if present.</p></li>
<li><p>If the <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> attribute is absent on <code class="docutils literal notranslate"><span class="pre">other</span></code> and
<code class="docutils literal notranslate"><span class="pre">other.__array_priority__</span> <span class="pre">></span> <span class="pre">self.__array_priority__</span></code>, <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a>
also returns <a class="reference external" href="https://docs.python.org/dev/library/constants.html#NotImplemented" title="(in Python v3.14)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">NotImplemented</span></code></a> (and the logic proceeds as in the
previous case). This ensures backwards compatibility with old versions
of NumPy.</p></li>
<li><p>Otherwise, <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a> unilaterally calls the corresponding Ufunc.
Ufuncs never return <code class="docutils literal notranslate"><span class="pre">NotImplemented</span></code>, so <strong>reflexive methods such
as</strong> <code class="docutils literal notranslate"><span class="pre">other.__rmul__</span></code> <strong>cannot be used to override arithmetic with
NumPy arrays if</strong> <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> <strong>is set</strong> to any value other than
<code class="docutils literal notranslate"><span class="pre">None</span></code>. Instead, their behavior needs to be changed by implementing
<code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> in a fashion consistent with the corresponding Ufunc,
e.g., <code class="docutils literal notranslate"><span class="pre">np.multiply</span></code>. See <a class="reference internal" href="#neps-ufunc-overrides-list-of-operators"><span class="std std-ref">List of operators and NumPy Ufuncs</span></a>
for a list of affected operators and their corresponding ufuncs.</p></li>
</ul>
<p>A class wishing to modify the interaction with <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a> in
binary operations therefore has two options:</p>
<ol class="arabic simple">
<li><p>Implement <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> and follow NumPy semantics for Python
binary operations (see below).</p></li>
<li><p>Set <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span> <span class="pre">=</span> <span class="pre">None</span></code>, and implement Python binary
operations freely. In this case, ufuncs called on this argument will
raise <a class="reference external" href="https://docs.python.org/dev/library/exceptions.html#TypeError" title="(in Python v3.14)"><code class="xref py py-exc docutils literal notranslate"><span class="pre">TypeError</span></code></a> (see
<a class="reference internal" href="#neps-ufunc-overrides-turning-ufuncs-off"><span class="std std-ref">Turning Ufuncs off</span></a>).</p></li>
</ol>
</section>
<section id="recommendations-for-implementing-binary-operations">
<h3>Recommendations for implementing binary operations<a class="headerlink" href="#recommendations-for-implementing-binary-operations" title="Link to this heading">#</a></h3>
<p>For most numerical classes, the easiest way to override binary
operations is thus to define <code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> and override the
corresponding Ufunc. The class can then, like <a class="reference external" href="https://numpy.org/devdocs/reference/generated/numpy.ndarray.html#numpy.ndarray" title="(in NumPy v2.3.dev0)"><code class="xref py py-class docutils literal notranslate"><span class="pre">ndarray</span></code></a> itself,
define the binary operators in terms of Ufuncs. Here, one has to take
some care to ensure that one allows for other classes to indicate they
are not compatible, i.e., implementations should be something like:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">def</span><span class="w"> </span><span class="nf">_disables_array_ufunc</span><span class="p">(</span><span class="n">obj</span><span class="p">):</span>
<span class="k">try</span><span class="p">:</span>
<span class="k">return</span> <span class="n">obj</span><span class="o">.</span><span class="n">__array_ufunc__</span> <span class="ow">is</span> <span class="kc">None</span>
<span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
<span class="k">return</span> <span class="kc">False</span>
<span class="k">class</span><span class="w"> </span><span class="nc">ArrayLike</span><span class="p">:</span>
<span class="o">...</span>
<span class="k">def</span><span class="w"> </span><span class="nf">__array_ufunc__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ufunc</span><span class="p">,</span> <span class="n">method</span><span class="p">,</span> <span class="o">*</span><span class="n">inputs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
<span class="o">...</span>
<span class="k">return</span> <span class="n">result</span>
<span class="c1"># Option 1: call ufunc directly</span>
<span class="k">def</span><span class="w"> </span><span class="fm">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
<span class="k">if</span> <span class="n">_disables_array_ufunc</span><span class="p">(</span><span class="n">other</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">NotImplemented</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">multiply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
<span class="k">def</span><span class="w"> </span><span class="fm">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
<span class="k">if</span> <span class="n">_disables_array_ufunc</span><span class="p">(</span><span class="n">other</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">NotImplemented</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">multiply</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
<span class="k">def</span><span class="w"> </span><span class="fm">__imul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">multiply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">out</span><span class="o">=</span><span class="p">(</span><span class="bp">self</span><span class="p">,))</span>
<span class="c1"># Option 2: call into one's own __array_ufunc__</span>
<span class="k">def</span><span class="w"> </span><span class="fm">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__array_ufunc__</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">multiply</span><span class="p">,</span> <span class="s1">'__call__'</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
<span class="k">def</span><span class="w"> </span><span class="fm">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__array_ufunc__</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">multiply</span><span class="p">,</span> <span class="s1">'__call__'</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
<span class="k">def</span><span class="w"> </span><span class="fm">__imul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
<span class="n">result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">__array_ufunc__</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">multiply</span><span class="p">,</span> <span class="s1">'__call__'</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span>
<span class="n">out</span><span class="o">=</span><span class="p">(</span><span class="bp">self</span><span class="p">,))</span>
<span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="bp">NotImplemented</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="o">...</span><span class="p">)</span>
</pre></div>
</div>
<p>To see why some care is necessary, consider another class <code class="docutils literal notranslate"><span class="pre">other</span></code> that
does not know how to deal with arrays and ufuncs, and thus has set
<code class="docutils literal notranslate"><span class="pre">__array_ufunc__</span></code> to <a class="reference external" href="https://docs.python.org/dev/library/constants.html#None" title="(in Python v3.14)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">None</span></code></a>, but does know how to do
multiplication:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">class</span><span class="w"> </span><span class="nc">MyObject</span><span class="p">:</span>
<span class="n">__array_ufunc__</span> <span class="o">=</span> <span class="kc">None</span>
<span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">value</span> <span class="o">=</span> <span class="n">value</span>
<span class="k">def</span><span class="w"> </span><span class="fm">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">return</span> <span class="s2">"MyObject(</span><span class="si">{!r}</span><span class="s2">)"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">value</span><span class="p">)</span>
<span class="k">def</span><span class="w"> </span><span class="fm">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
<span class="k">return</span> <span class="n">MyObject</span><span class="p">(</span><span class="mi">1234</span><span class="p">)</span>
<span class="k">def</span><span class="w"> </span><span class="fm">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
<span class="k">return</span> <span class="n">MyObject</span><span class="p">(</span><span class="mi">4321</span><span class="p">)</span>
</pre></div>
</div>
<p>For either option above, we get the expected result:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">mine</span> <span class="o">=</span> <span class="n">MyObject</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">arr</span> <span class="o">=</span> <span class="n">ArrayLike</span><span class="p">([</span><span class="mi">0</span><span class="p">])</span>
<span class="n">mine</span> <span class="o">*</span> <span class="n">arr</span> <span class="c1"># -> MyObject(1234)</span>
<span class="n">mine</span> <span class="o">*=</span> <span class="n">arr</span> <span class="c1"># -> MyObject(1234)</span>
<span class="n">arr</span> <span class="o">*</span> <span class="n">mine</span> <span class="c1"># -> MyObject(4321)</span>