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<section id="nep-20-expansion-of-generalized-universal-function-signatures">
<span id="nep20"></span><h1>NEP 20 — Expansion of generalized universal function signatures<a class="headerlink" href="#nep-20-expansion-of-generalized-universal-function-signatures" 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>Marten van Kerkwijk <<a class="reference external" href="mailto:mhvk%40astro.utoronto.ca">mhvk<span>@</span>astro<span>.</span>utoronto<span>.</span>ca</a>></p>
</dd>
<dt class="field-even">Status<span class="colon">:</span></dt>
<dd class="field-even"><p>Final</p>
</dd>
<dt class="field-odd">Type<span class="colon">:</span></dt>
<dd class="field-odd"><p>Standards Track</p>
</dd>
<dt class="field-even">Created<span class="colon">:</span></dt>
<dd class="field-even"><p>2018-06-10</p>
</dd>
<dt class="field-odd">Resolution<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference external" href="https://mail.python.org/pipermail/numpy-discussion/2018-April/077959.html">https://mail.python.org/pipermail/numpy-discussion/2018-April/077959.html</a>,
<a class="reference external" href="https://mail.python.org/pipermail/numpy-discussion/2018-May/078078.html">https://mail.python.org/pipermail/numpy-discussion/2018-May/078078.html</a></p>
</dd>
</dl>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The proposal to add fixed (i) and flexible (ii) dimensions
was accepted, while that to add broadcastable (iii) ones was deferred.</p>
</div>
<section id="abstract">
<h2>Abstract<a class="headerlink" href="#abstract" title="Link to this heading">#</a></h2>
<p>Generalized universal functions are, as their name indicates, generalization
of universal functions: they operate on non-scalar elements. Their signature
describes the structure of the elements they operate on, with names linking
dimensions of the operands that should be the same. Here, it is proposed to
extend the signature to allow the signature to indicate that a dimension (i)
has fixed size; (ii) can be absent; and (iii) can be broadcast.</p>
</section>
<section id="detailed-description">
<h2>Detailed description<a class="headerlink" href="#detailed-description" title="Link to this heading">#</a></h2>
<p>Each part of the proposal is driven by specific needs <a class="footnote-reference brackets" href="#id12" id="id1" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a>.</p>
<ol class="arabic">
<li><p>Fixed-size dimensions. Code working with spatial vectors often explicitly
is for 2 or 3-dimensional space (e.g., the code from the <a class="reference external" href="http://www.iausofa.org/">Standards Of
Fundamental Astronomy</a>, which the author hopes
to wrap using gufuncs for astropy <a class="footnote-reference brackets" href="#id13" id="id2" role="doc-noteref"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></a>). The signature should be able to
indicate that. E.g., the signature of a function that converts a polar
angle to a two-dimensional cartesian unit vector would currently have to be
<code class="docutils literal notranslate"><span class="pre">()->(n)</span></code>, with there being no way to indicate that <code class="docutils literal notranslate"><span class="pre">n</span></code> has to equal 2.
Indeed, this signature is particularly annoying since without putting in an
output argument, the current gufunc wrapper code fails because it cannot
determine <code class="docutils literal notranslate"><span class="pre">n</span></code>. Similarly, the signature for an cross product of two
3-dimensional vectors has to be <code class="docutils literal notranslate"><span class="pre">(n),(n)->(n)</span></code>, with again no way to
indicate that <code class="docutils literal notranslate"><span class="pre">n</span></code> has to equal 3. Hence, the proposal here to allow one
to give numerical values in addition to variable names. Thus, angle to
two-dimensional unit vector would be <code class="docutils literal notranslate"><span class="pre">()->(2)</span></code>; two angles to
three-dimensional unit vector <code class="docutils literal notranslate"><span class="pre">(),()->(3)</span></code>; and that for the cross
product of two three-dimensional vectors would be <code class="docutils literal notranslate"><span class="pre">(3),(3)->(3)</span></code>.</p></li>
<li><p>Possibly missing dimensions. This part is almost entirely driven by the
wish to wrap <code class="docutils literal notranslate"><span class="pre">matmul</span></code> in a gufunc. <code class="docutils literal notranslate"><span class="pre">matmul</span></code> stands for matrix
multiplication, and if it did only that, it could be covered with the
signature <code class="docutils literal notranslate"><span class="pre">(m,n),(n,p)->(m,p)</span></code>. However, it has special cases for when a
dimension is missing, allowing either argument to be treated as a single
vector, with the function thus becoming, effectively, vector-matrix,
matrix-vector, or vector-vector multiplication (but with no
broadcasting). To support this, it is suggested to allow postfixing a
dimension name with a question mark to indicate that the dimension does not
necessarily have to be present.</p>
<p>With this addition, the signature for <code class="docutils literal notranslate"><span class="pre">matmul</span></code> can be expressed as
<code class="docutils literal notranslate"><span class="pre">(m?,n),(n,p?)->(m?,p?)</span></code>. This indicates that if, e.g., the second
operand has only one dimension, for the purposes of the elementary function
it will be treated as if that input has core shape <code class="docutils literal notranslate"><span class="pre">(n,</span> <span class="pre">1)</span></code>, and the
output has the corresponding core shape of <code class="docutils literal notranslate"><span class="pre">(m,</span> <span class="pre">1)</span></code>. The actual output
array, however, has the flexible dimension removed, i.e., it will have
shape <code class="docutils literal notranslate"><span class="pre">(...,</span> <span class="pre">m)</span></code>. Similarly, if both arguments have only a single
dimension, the inputs will be presented as having shapes <code class="docutils literal notranslate"><span class="pre">(1,</span> <span class="pre">n)</span></code> and
<code class="docutils literal notranslate"><span class="pre">(n,</span> <span class="pre">1)</span></code> to the elementary function, and the output as <code class="docutils literal notranslate"><span class="pre">(1,</span> <span class="pre">1)</span></code>, while
the actual output array returned will have shape <code class="docutils literal notranslate"><span class="pre">()</span></code>. In this way, the
signature allows one to use a single elementary function for four related
but different signatures, <code class="docutils literal notranslate"><span class="pre">(m,n),(n,p)->(m,p)</span></code>, <code class="docutils literal notranslate"><span class="pre">(n),(n,p)->(p)</span></code>,
<code class="docutils literal notranslate"><span class="pre">(m,n),(n)->(m)</span></code> and <code class="docutils literal notranslate"><span class="pre">(n),(n)->()</span></code>.</p>
</li>
<li><p>Dimensions that can be broadcast. For some applications, broadcasting
between operands makes sense. For instance, an <code class="docutils literal notranslate"><span class="pre">all_equal</span></code> function that
compares vectors in arrays could have a signature <code class="docutils literal notranslate"><span class="pre">(n),(n)->()</span></code>, but this
forces both operands to be arrays, while it would be useful also to check
that, e.g., all parts of a vector are constant (maybe zero). The proposal
is to allow the implementer of a gufunc to indicate that a dimension can be
broadcast by post-fixing the dimension name with <code class="docutils literal notranslate"><span class="pre">|1</span></code>. Hence, the
signature for <code class="docutils literal notranslate"><span class="pre">all_equal</span></code> would become <code class="docutils literal notranslate"><span class="pre">(n|1),(n|1)->()</span></code>. The
signature seems handy more generally for “chained ufuncs”; e.g., another
application might be in a putative ufunc implementing <code class="docutils literal notranslate"><span class="pre">sumproduct</span></code>.</p>
<p>Another example that arose in the discussion, is of a weighted mean, which
might look like <code class="docutils literal notranslate"><span class="pre">weighted_mean(y,</span> <span class="pre">sigma[,</span> <span class="pre">axis,</span> <span class="pre">...])</span></code>, returning the
mean and its uncertainty. With a signature of <code class="docutils literal notranslate"><span class="pre">(n),(n)->(),()</span></code>, one
would be forced to always give as many sigmas as there are data points,
while broadcasting would allow one to give a single sigma for all points
(which is still useful to calculate the uncertainty on the mean).</p>
</li>
</ol>
</section>
<section id="implementation">
<h2>Implementation<a class="headerlink" href="#implementation" title="Link to this heading">#</a></h2>
<p>The proposed changes have all been implemented <a class="footnote-reference brackets" href="#id15" id="id3" role="doc-noteref"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></a>, <a class="footnote-reference brackets" href="#id16" id="id4" role="doc-noteref"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></a>, <a class="footnote-reference brackets" href="#id17" id="id5" role="doc-noteref"><span class="fn-bracket">[</span>5<span class="fn-bracket">]</span></a>. These PRs
extend the ufunc structure with two new fields, each of size equal to the
number of distinct dimensions, with <code class="docutils literal notranslate"><span class="pre">core_dim_sizes</span></code> holding possibly fixed
sizes, and <code class="docutils literal notranslate"><span class="pre">core_dim_flags</span></code> holding flags indicating whether a dimension can
be missing or broadcast. To ensure we can distinguish between this new
version and previous versions, an unused entry <code class="docutils literal notranslate"><span class="pre">reserved1</span></code> is repurposed as
a version number.</p>
<p>In the implementation, care is taken that to the elementary function flagged
dimensions are not treated any differently than non-flagged ones: for
instance, sizes of fixed-size dimensions are still passed on to the elementary
function (but the loop can now count on that size being equal to the fixed one
given in the signature).</p>
<p>An implementation detail to be decided upon is whether it might be handy to
have a summary of all flags. This could possibly be stored in <code class="docutils literal notranslate"><span class="pre">core_enabled</span></code>
(which currently is a bool), with non-zero continuing to indicate a gufunc,
but specific flags indicating whether or not a gufunc uses fixed, flexible, or
broadcastable dimensions.</p>
<p>With the above, the formal definition of the syntax would become <a class="footnote-reference brackets" href="#id16" id="id6" role="doc-noteref"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></a>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o"><</span><span class="n">Signature</span><span class="o">></span> <span class="p">:</span><span class="o">:=</span> <span class="o"><</span><span class="n">Input</span> <span class="n">arguments</span><span class="o">></span> <span class="s2">"->"</span> <span class="o"><</span><span class="n">Output</span> <span class="n">arguments</span><span class="o">></span>
<span class="o"><</span><span class="n">Input</span> <span class="n">arguments</span><span class="o">></span> <span class="p">:</span><span class="o">:=</span> <span class="o"><</span><span class="n">Argument</span> <span class="nb">list</span><span class="o">></span>
<span class="o"><</span><span class="n">Output</span> <span class="n">arguments</span><span class="o">></span> <span class="p">:</span><span class="o">:=</span> <span class="o"><</span><span class="n">Argument</span> <span class="nb">list</span><span class="o">></span>
<span class="o"><</span><span class="n">Argument</span> <span class="nb">list</span><span class="o">></span> <span class="p">:</span><span class="o">:=</span> <span class="n">nil</span> <span class="o">|</span> <span class="o"><</span><span class="n">Argument</span><span class="o">></span> <span class="o">|</span> <span class="o"><</span><span class="n">Argument</span><span class="o">></span> <span class="s2">","</span> <span class="o"><</span><span class="n">Argument</span> <span class="nb">list</span><span class="o">></span>
<span class="o"><</span><span class="n">Argument</span><span class="o">></span> <span class="p">:</span><span class="o">:=</span> <span class="s2">"("</span> <span class="o"><</span><span class="n">Core</span> <span class="n">dimension</span> <span class="nb">list</span><span class="o">></span> <span class="s2">")"</span>
<span class="o"><</span><span class="n">Core</span> <span class="n">dimension</span> <span class="nb">list</span><span class="o">></span> <span class="p">:</span><span class="o">:=</span> <span class="n">nil</span> <span class="o">|</span> <span class="o"><</span><span class="n">Core</span> <span class="n">dimension</span><span class="o">></span> <span class="o">|</span>
<span class="o"><</span><span class="n">Core</span> <span class="n">dimension</span><span class="o">></span> <span class="s2">","</span> <span class="o"><</span><span class="n">Core</span> <span class="n">dimension</span> <span class="nb">list</span><span class="o">></span>
<span class="o"><</span><span class="n">Core</span> <span class="n">dimension</span><span class="o">></span> <span class="p">:</span><span class="o">:=</span> <span class="o"><</span><span class="n">Dimension</span> <span class="n">name</span><span class="o">></span> <span class="o"><</span><span class="n">Dimension</span> <span class="n">modifier</span><span class="o">></span>
<span class="o"><</span><span class="n">Dimension</span> <span class="n">name</span><span class="o">></span> <span class="p">:</span><span class="o">:=</span> <span class="n">valid</span> <span class="n">Python</span> <span class="n">variable</span> <span class="n">name</span> <span class="o">|</span> <span class="n">valid</span> <span class="n">integer</span>
<span class="o"><</span><span class="n">Dimension</span> <span class="n">modifier</span><span class="o">></span> <span class="p">:</span><span class="o">:=</span> <span class="n">nil</span> <span class="o">|</span> <span class="s2">"|1"</span> <span class="o">|</span> <span class="s2">"?"</span>
</pre></div>
</div>
<ol class="arabic simple">
<li><p>All quotes are for clarity.</p></li>
<li><p>Unmodified core dimensions that share the same name must have the same size.
Each dimension name typically corresponds to one level of looping in the
elementary function’s implementation.</p></li>
<li><p>White spaces are ignored.</p></li>
<li><p>An integer as a dimension name freezes that dimension to the value.</p></li>
<li><p>If a name if suffixed with the <code class="docutils literal notranslate"><span class="pre">|1</span></code> modifier, it is allowed to broadcast
against other dimensions with the same name. All input dimensions
must share this modifier, while no output dimensions should have it.</p></li>
<li><p>If the name is suffixed with the <code class="docutils literal notranslate"><span class="pre">?</span></code> modifier, the dimension is a core
dimension only if it exists on all inputs and outputs that share it;
otherwise it is ignored (and replaced by a dimension of size 1 for the
elementary function).</p></li>
</ol>
<p>Examples of signatures <a class="footnote-reference brackets" href="#id16" id="id7" role="doc-noteref"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></a>:</p>
<div class="pst-scrollable-table-container"><table class="table">
<tbody>
<tr class="row-odd"><td><p>Signature</p></td>
<td><p>Possible use</p></td>
</tr>
<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">(),()->()</span></code></p></td>
<td><p>Addition</p></td>
</tr>
<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">(i)->()</span></code></p></td>
<td><p>Sum over last axis</p></td>
</tr>
<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">(i|1),(i|1)->()</span></code></p></td>
<td><p>Test for equality along axis,
allowing comparison with a scalar</p></td>
</tr>
<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">(i),(i)->()</span></code></p></td>
<td><p>inner vector product</p></td>
</tr>
<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">(m,n),(n,p)->(m,p)</span></code></p></td>
<td><p>matrix multiplication</p></td>
</tr>
<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">(n),(n,p)->(p)</span></code></p></td>
<td><p>vector-matrix multiplication</p></td>
</tr>
<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">(m,n),(n)->(m)</span></code></p></td>
<td><p>matrix-vector multiplication</p></td>
</tr>
<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">(m?,n),(n,p?)->(m?,p?)</span></code></p></td>
<td><p>all four of the above at once,
except vectors cannot have loop
dimensions (ie, like <code class="docutils literal notranslate"><span class="pre">matmul</span></code>)</p></td>
</tr>
<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">(3),(3)->(3)</span></code></p></td>
<td><p>cross product for 3-vectors</p></td>
</tr>
<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">(i,t),(j,t)->(i,j)</span></code></p></td>
<td><p>inner over the last dimension,
outer over the second to last,
and loop/broadcast over the rest.</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="backward-compatibility">
<h2>Backward compatibility<a class="headerlink" href="#backward-compatibility" title="Link to this heading">#</a></h2>
<p>One possible worry is the change in ufunc structure. For most applications,
which call <code class="docutils literal notranslate"><span class="pre">PyUFunc_FromDataAndSignature</span></code>, this is entirely transparent.
Furthermore, by repurposing <code class="docutils literal notranslate"><span class="pre">reserved1</span></code> as a version number, code compiled
against older versions of numpy will continue to work (though one will get a
warning upon import of that code with a newer version of numpy), except if
code explicitly changes the <code class="docutils literal notranslate"><span class="pre">reserved1</span></code> entry.</p>
</section>
<section id="alternatives">
<h2>Alternatives<a class="headerlink" href="#alternatives" title="Link to this heading">#</a></h2>
<p>It was suggested instead of extending the signature, to have multiple
dispatch, so that, e.g., <code class="docutils literal notranslate"><span class="pre">matmul</span></code> would simply have the multiple signatures
it supports, i.e., instead of <code class="docutils literal notranslate"><span class="pre">(m?,n),(n,p?)->(m?,p?)</span></code> one would have
<code class="docutils literal notranslate"><span class="pre">(m,n),(n,p)->(m,p)</span> <span class="pre">|</span> <span class="pre">(n),(n,p)->(p)</span> <span class="pre">|</span> <span class="pre">(m,n),(n)->(m)</span> <span class="pre">|</span> <span class="pre">(n),(n)->()</span></code>. A
disadvantage of this is that the developer now has to make sure that the
elementary function can deal with these different signatures. Furthermore,
the expansion quickly becomes cumbersome. For instance, for the <code class="docutils literal notranslate"><span class="pre">all_equal</span></code>
signature of <code class="docutils literal notranslate"><span class="pre">(n|1),(n|1)->()</span></code>, one would have to have five entries:
<code class="docutils literal notranslate"><span class="pre">(n),(n)->()</span> <span class="pre">|</span> <span class="pre">(n),(1)->()</span> <span class="pre">|</span> <span class="pre">(1),(n)->()</span> <span class="pre">|</span> <span class="pre">(n),()->()</span> <span class="pre">|</span> <span class="pre">(),(n)->()</span></code>. For
signatures like <code class="docutils literal notranslate"><span class="pre">(m|1,n|1,o|1),(m|1,n|1,o|1)->()</span></code> (from the <code class="docutils literal notranslate"><span class="pre">cube_equal</span></code>
test case in <a class="footnote-reference brackets" href="#id16" id="id8" role="doc-noteref"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></a>), it is not even worth writing out the expansion.</p>
<p>For broadcasting, the alternative suffix of <code class="docutils literal notranslate"><span class="pre">^</span></code> was suggested (as
broadcasting can be thought of as increasing the size of the array). This
seems less clear. Furthermore, it was wondered whether it should not just be
an all-or-nothing flag. This could be the case, though given the postfix
for flexible dimensions, arguably another postfix is clearer (as is the
implementation).</p>
</section>
<section id="discussion">
<h2>Discussion<a class="headerlink" href="#discussion" title="Link to this heading">#</a></h2>
<p>The proposals here were discussed at fair length on the mailing list <a class="footnote-reference brackets" href="#id18" id="id9" role="doc-noteref"><span class="fn-bracket">[</span>6<span class="fn-bracket">]</span></a>,
<a class="footnote-reference brackets" href="#id19" id="id10" role="doc-noteref"><span class="fn-bracket">[</span>7<span class="fn-bracket">]</span></a>. The main points of contention were whether the use cases were
sufficiently strong. In particular, for frozen dimensions, it was argued that
checks on the right number could be put in loop selection code. This seems
much less clear for no benefit.</p>
<p>For broadcasting, the lack of examples of elementary functions that might need
it was noted, with it being questioned whether something like <code class="docutils literal notranslate"><span class="pre">all_equal</span></code>
was best done with a gufunc rather than as a special method on <code class="docutils literal notranslate"><span class="pre">np.equal</span></code>.
One counter-argument to this would be that there is an actual PR for
<code class="docutils literal notranslate"><span class="pre">all_equal</span></code> <a class="footnote-reference brackets" href="#id20" id="id11" role="doc-noteref"><span class="fn-bracket">[</span>8<span class="fn-bracket">]</span></a>. Another that even if one were to use a method, it would
be good to be able to express their signature (just as is possible at least
for <code class="docutils literal notranslate"><span class="pre">reduce</span></code> and <code class="docutils literal notranslate"><span class="pre">accumulate</span></code>).</p>
<p>A final argument was that we were making the gufuncs too complex. This
arguably holds for the dimensions that can be omitted, but that also has the
strongest use case. The frozen dimensions has a very simple implementation and
its meaning is obvious. The ability to broadcast is simple too, once the
flexible dimensions are supported.</p>
</section>
<section id="references-and-footnotes">
<h2>References and footnotes<a class="headerlink" href="#references-and-footnotes" title="Link to this heading">#</a></h2>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id12" 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>Identified needs and suggestions for the implementation are not all by
the author. In particular, the suggestion for fixed dimensions and
initial implementation was by Jaime Frio (<a class="reference external" href="https://github.com/numpy/numpy/pull/5015">gh-5015</a>), the suggestion of <code class="docutils literal notranslate"><span class="pre">?</span></code>
to indicate dimensions can be omitted was by Nathaniel Smith, and the
initial implementation of that by Matti Picus (<a class="reference external" href="https://github.com/numpy/numpy/pull/11132">gh-11132</a>).</p>
</aside>
<aside class="footnote brackets" id="id13" 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="reference external" href="https://github.com/astropy/astropy/pull/7502">wrap ERFA functions in gufuncs</a> (<a class="reference external" href="https://github.com/liberfa/erfa">ERFA</a>) is the less stringently licensed
version of <a class="reference external" href="http://www.iausofa.org/">Standards Of Fundamental Astronomy</a></p>
</aside>
<aside class="footnote brackets" id="id15" 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="reference external" href="https://github.com/numpy/numpy/pull/11175">fixed-size and flexible dimensions</a></p>
</aside>
<aside class="footnote brackets" id="id16" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id4">1</a>,<a role="doc-backlink" href="#id6">2</a>,<a role="doc-backlink" href="#id7">3</a>,<a role="doc-backlink" href="#id8">4</a>)</span>
<p><a class="reference external" href="https://github.com/numpy/numpy/pull/11179">broadcastable dimensions</a></p>
</aside>
<aside class="footnote brackets" id="id17" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id5">5</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://github.com/numpy/numpy/pull/11133">use in matmul</a></p>
</aside>
<aside class="footnote brackets" id="id18" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id9">6</a><span class="fn-bracket">]</span></span>
<p>Discusses implementations for <code class="docutils literal notranslate"><span class="pre">matmul</span></code>:
<a class="reference external" href="https://mail.python.org/pipermail/numpy-discussion/2018-May/077972.html">https://mail.python.org/pipermail/numpy-discussion/2018-May/077972.html</a>,
<a class="reference external" href="https://mail.python.org/pipermail/numpy-discussion/2018-May/078021.html">https://mail.python.org/pipermail/numpy-discussion/2018-May/078021.html</a></p>
</aside>
<aside class="footnote brackets" id="id19" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id10">7</a><span class="fn-bracket">]</span></span>
<p>Broadcasting:
<a class="reference external" href="https://mail.python.org/pipermail/numpy-discussion/2018-May/078078.html">https://mail.python.org/pipermail/numpy-discussion/2018-May/078078.html</a></p>
</aside>
<aside class="footnote brackets" id="id20" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id11">8</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://github.com/numpy/numpy/pull/8528">Logical gufuncs</a> (includes
<code class="docutils literal notranslate"><span class="pre">all_equal</span></code>)</p>
</aside>
</aside>
</section>
<section id="copyright">
<h2>Copyright<a class="headerlink" href="#copyright" title="Link to this heading">#</a></h2>
<p>This document has been placed in the public domain.</p>
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