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<section id="nep-27-zero-rank-arrays">
<span id="nep27"></span><h1>NEP 27 — Zero rank arrays<a class="headerlink" href="#nep-27-zero-rank-arrays" 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>Alexander Belopolsky (sasha), transcribed Matt Picus <<a class="reference external" href="mailto:matti.picus%40gmail.com">matti<span>.</span>picus<span>@</span>gmail<span>.</span>com</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>Informational</p>
</dd>
<dt class="field-even">Created<span class="colon">:</span></dt>
<dd class="field-even"><p>2006-06-10</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-October/078824.html">https://mail.python.org/pipermail/numpy-discussion/2018-October/078824.html</a></p>
</dd>
</dl>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>NumPy has both zero rank arrays and scalars. This design document, adapted
from a <a class="reference external" href="https://web.archive.org/web/20100503065506/http://projects.scipy.org:80/numpy/wiki/ZeroRankArray">2006 wiki entry</a>, describes what zero rank arrays are and why they
exist. It was transcribed 2018-10-13 into a NEP and links were updated.
The pull request sparked <a class="reference external" href="https://github.com/numpy/numpy/pull/12166">a lively discussion</a> about the continued need
for zero rank arrays and scalars in NumPy.</p>
<p>Some of the information here is dated, for instance indexing of 0-D arrays
now is now implemented and does not error.</p>
</div>
<section id="zero-rank-arrays">
<h2>Zero-rank arrays<a class="headerlink" href="#zero-rank-arrays" title="Link to this heading">#</a></h2>
<p>Zero-rank arrays are arrays with shape=(). For example:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">x</span><span class="o">.</span><span class="n">shape</span>
<span class="go">()</span>
</pre></div>
</div>
</section>
<section id="zero-rank-arrays-and-array-scalars">
<h2>Zero-rank arrays and array scalars<a class="headerlink" href="#zero-rank-arrays-and-array-scalars" title="Link to this heading">#</a></h2>
<p>Array scalars are similar to zero-rank arrays in many aspects:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">int_</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">shape</span>
<span class="go">()</span>
</pre></div>
</div>
<p>They even print the same:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="nb">print</span> <span class="n">int_</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">1</span>
<span class="gp">>>> </span><span class="nb">print</span> <span class="n">array</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">1</span>
</pre></div>
</div>
<p>However there are some important differences:</p>
<ul class="simple">
<li><p>Array scalars are immutable</p></li>
<li><p>Array scalars have different python type for different data types</p></li>
</ul>
</section>
<section id="motivation-for-array-scalars">
<h2>Motivation for array scalars<a class="headerlink" href="#motivation-for-array-scalars" title="Link to this heading">#</a></h2>
<p>NumPy’s design decision to provide 0-d arrays and array scalars in addition to
native python types goes against one of the fundamental python design
principles that there should be only one obvious way to do it. In this section
we will try to explain why it is necessary to have three different ways to
represent a number.</p>
<p>There were several numpy-discussion threads:</p>
<ul class="simple">
<li><p><a class="reference external" href="https://mail.python.org/pipermail/numpy-discussion/2002-September/001600.html">rank-0 arrays</a> in a 2002 mailing list thread.</p></li>
<li><p>Thoughts about zero dimensional arrays vs Python scalars in a <a class="reference external" href="https://sourceforge.net/p/numpy/mailman/message/11299166">2005 mailing list thread</a>]</p></li>
</ul>
<p>It has been suggested several times that NumPy just use rank-0 arrays to
represent scalar quantities in all case. Pros and cons of converting rank-0
arrays to scalars were summarized as follows:</p>
<ul class="simple">
<li><p>Pros:</p>
<ul>
<li><p>Some cases when Python expects an integer (the most
dramatic is when slicing and indexing a sequence:
_PyEval_SliceIndex in ceval.c) it will not try to
convert it to an integer first before raising an error.
Therefore it is convenient to have 0-dim arrays that
are integers converted for you by the array object.</p></li>
<li><p>No risk of user confusion by having two types that
are nearly but not exactly the same and whose separate
existence can only be explained by the history of
Python and NumPy development.</p></li>
<li><p>No problems with code that does explicit typechecks
<code class="docutils literal notranslate"><span class="pre">(isinstance(x,</span> <span class="pre">float)</span></code> or <code class="docutils literal notranslate"><span class="pre">type(x)</span> <span class="pre">==</span> <span class="pre">types.FloatType)</span></code>. Although
explicit typechecks are considered bad practice in general, there are a
couple of valid reasons to use them.</p></li>
<li><p>No creation of a dependency on Numeric in pickle
files (though this could also be done by a special case
in the pickling code for arrays)</p></li>
</ul>
</li>
<li><p>Cons:</p>
<ul>
<li><p>It is difficult to write generic code because scalars
do not have the same methods and attributes as arrays.
(such as <code class="docutils literal notranslate"><span class="pre">.type</span></code> or <code class="docutils literal notranslate"><span class="pre">.shape</span></code>). Also Python scalars have
different numeric behavior as well.</p></li>
<li><p>This results in a special-case checking that is not
pleasant. Fundamentally it lets the user believe that
somehow multidimensional homogeneous arrays
are something like Python lists (which except for
Object arrays they are not).</p></li>
</ul>
</li>
</ul>
<p>NumPy implements a solution that is designed to have all the pros and none of the cons above.</p>
<blockquote>
<div><p>Create Python scalar types for all of the 21 types and also
inherit from the three that already exist. Define equivalent
methods and attributes for these Python scalar types.</p>
</div></blockquote>
</section>
<section id="the-need-for-zero-rank-arrays">
<h2>The need for zero-rank arrays<a class="headerlink" href="#the-need-for-zero-rank-arrays" title="Link to this heading">#</a></h2>
<p>Once the idea to use zero-rank arrays to represent scalars was rejected, it was
natural to consider whether zero-rank arrays can be eliminated altogether.
However there are some important use cases where zero-rank arrays cannot be
replaced by array scalars. See also <a class="reference external" href="https://mail.python.org/pipermail/numpy-discussion/2006-February/006384.html">A case for rank-0 arrays</a> from February
2006.</p>
<ul>
<li><p>Output arguments:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>>>> y = int_(5)
>>> add(5,5,x)
array(10)
>>> x
array(10)
>>> add(5,5,y)
Traceback (most recent call last):
File "<stdin>", line 1, in ?
TypeError: return arrays must be of ArrayType
</pre></div>
</div>
</li>
<li><p>Shared data:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="mi">2</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">.</span><span class="n">shape</span> <span class="o">=</span> <span class="p">()</span>
<span class="gp">>>> </span><span class="n">y</span>
<span class="go">array(2)</span>
<span class="gp">>>> </span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">20</span>
<span class="gp">>>> </span><span class="n">y</span>
<span class="go">array(20)</span>
</pre></div>
</div>
</li>
</ul>
</section>
<section id="indexing-of-zero-rank-arrays">
<h2>Indexing of zero-rank arrays<a class="headerlink" href="#indexing-of-zero-rank-arrays" title="Link to this heading">#</a></h2>
<p>As of NumPy release 0.9.3, zero-rank arrays do not support any indexing:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span><span class="p">[</span><span class="o">...</span><span class="p">]</span>
<span class="gt">Traceback (most recent call last):</span>
File <span class="nb">"<stdin>"</span>, line <span class="m">1</span>, in <span class="n">?</span>
<span class="gr">IndexError</span>: <span class="n">0-d arrays can't be indexed.</span>
</pre></div>
</div>
<p>On the other hand there are several cases that make sense for rank-zero arrays.</p>
<section id="ellipsis-and-empty-tuple">
<h3>Ellipsis and empty tuple<a class="headerlink" href="#ellipsis-and-empty-tuple" title="Link to this heading">#</a></h3>
<p>Alexander started a <a class="reference external" href="https://mail.python.org/pipermail/numpy-discussion/2006-January/005579.html">Jan 2006 discussion</a> on scipy-dev
with the following proposal:</p>
<blockquote>
<div><p>… it may be reasonable to allow <code class="docutils literal notranslate"><span class="pre">a[...]</span></code>. This way
ellipsis can be interpreted as any number of <code class="docutils literal notranslate"><span class="pre">:</span></code> s including zero.
Another subscript operation that makes sense for scalars would be
<code class="docutils literal notranslate"><span class="pre">a[...,newaxis]</span></code> or even <code class="docutils literal notranslate"><span class="pre">a[{newaxis,</span> <span class="pre">}*</span> <span class="pre">...,</span> <span class="pre">{newaxis,}*]</span></code>, where
<code class="docutils literal notranslate"><span class="pre">{newaxis,}*</span></code> stands for any number of comma-separated newaxis tokens.
This will allow one to use ellipsis in generic code that would work on
any numpy type.</p>
</div></blockquote>
<p>Francesc Altet supported the idea of <code class="docutils literal notranslate"><span class="pre">[...]</span></code> on zero-rank arrays and
<a class="reference external" href="https://mail.python.org/pipermail/numpy-discussion/2006-January/005572.html">suggested</a> that <code class="docutils literal notranslate"><span class="pre">[()]</span></code> be supported as well.</p>
<p>Francesc’s proposal was:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">In</span> <span class="p">[</span><span class="mi">65</span><span class="p">]:</span> <span class="nb">type</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="mi">0</span><span class="p">)[</span><span class="o">...</span><span class="p">])</span>
<span class="n">Out</span><span class="p">[</span><span class="mi">65</span><span class="p">]:</span> <span class="o"><</span><span class="nb">type</span> <span class="s1">'numpy.ndarray'</span><span class="o">></span>
<span class="n">In</span> <span class="p">[</span><span class="mi">66</span><span class="p">]:</span> <span class="nb">type</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="mi">0</span><span class="p">)[()])</span> <span class="c1"># Indexing a la numarray</span>
<span class="n">Out</span><span class="p">[</span><span class="mi">66</span><span class="p">]:</span> <span class="o"><</span><span class="nb">type</span> <span class="s1">'int32_arrtype'</span><span class="o">></span>
<span class="n">In</span> <span class="p">[</span><span class="mi">67</span><span class="p">]:</span> <span class="nb">type</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">item</span><span class="p">())</span> <span class="c1"># already works</span>
<span class="n">Out</span><span class="p">[</span><span class="mi">67</span><span class="p">]:</span> <span class="o"><</span><span class="nb">type</span> <span class="s1">'int'</span><span class="o">></span>
</pre></div>
</div>
<p>There is a consensus that for a zero-rank array <code class="docutils literal notranslate"><span class="pre">x</span></code>, both <code class="docutils literal notranslate"><span class="pre">x[...]</span></code> and <code class="docutils literal notranslate"><span class="pre">x[()]</span></code> should be valid, but the question
remains on what should be the type of the result - zero rank ndarray or <code class="docutils literal notranslate"><span class="pre">x.dtype</span></code>?</p>
<dl>
<dt>(Alexander)</dt><dd><p>First, whatever choice is made for <code class="docutils literal notranslate"><span class="pre">x[...]</span></code> and <code class="docutils literal notranslate"><span class="pre">x[()]</span></code> they should be
the same because <code class="docutils literal notranslate"><span class="pre">...</span></code> is just syntactic sugar for “as many <cite>:</cite> as
necessary”, which in the case of zero rank leads to <code class="docutils literal notranslate"><span class="pre">...</span> <span class="pre">=</span> <span class="pre">(:,)*0</span> <span class="pre">=</span> <span class="pre">()</span></code>.
Second, rank zero arrays and numpy scalar types are interchangeable within
numpy, but numpy scalars can be use in some python constructs where ndarrays
can’t. For example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </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="gt">Traceback (most recent call last):</span>
File <span class="nb">"<stdin>"</span>, line <span class="m">1</span>, in <span class="n">?</span>
<span class="gr">TypeError</span>: <span class="n">tuple indices must be integers</span>
<span class="gp">>>> </span><span class="p">(</span><span class="mi">1</span><span class="p">,)[</span><span class="n">int32</span><span class="p">(</span><span class="mi">0</span><span class="p">)]</span>
<span class="go">1</span>
</pre></div>
</div>
</dd>
</dl>
<p>Since most if not all numpy function automatically convert zero-rank arrays to scalars on return, there is no reason for
<code class="docutils literal notranslate"><span class="pre">[...]</span></code> and <code class="docutils literal notranslate"><span class="pre">[()]</span></code> operations to be different.</p>
<p>See SVN changeset 1864 (which became git commit <a class="reference external" href="https://github.com/numpy/numpy/commit/9024ff0dc052888b5922dde0f3e615607a9e99d7">9024ff0</a>) for
implementation of <code class="docutils literal notranslate"><span class="pre">x[...]</span></code> and <code class="docutils literal notranslate"><span class="pre">x[()]</span></code> returning numpy scalars.</p>
<p>See SVN changeset 1866 (which became git commit <a class="reference external" href="https://github.com/numpy/numpy/commit/743d922bf5893acf00ac92e823fe12f460726f90">743d922</a>) for
implementation of <code class="docutils literal notranslate"><span class="pre">x[...]</span> <span class="pre">=</span> <span class="pre">v</span></code> and <code class="docutils literal notranslate"><span class="pre">x[()]</span> <span class="pre">=</span> <span class="pre">v</span></code></p>
</section>
<section id="increasing-rank-with-newaxis">
<h3>Increasing rank with newaxis<a class="headerlink" href="#increasing-rank-with-newaxis" title="Link to this heading">#</a></h3>
<p>Everyone who commented liked this feature, so as of SVN changeset 1871 (which became git commit <a class="reference external" href="https://github.com/numpy/numpy/commit/b32744e3fc5b40bdfbd626dcc1f72907d77c01c4">b32744e</a>) any number of ellipses and
newaxis tokens can be placed as a subscript argument for a zero-rank array. For
example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">x</span><span class="p">[</span><span class="n">newaxis</span><span class="p">,</span><span class="o">...</span><span class="p">,</span><span class="n">newaxis</span><span class="p">,</span><span class="o">...</span><span class="p">]</span>
<span class="go">array([[1]])</span>
</pre></div>
</div>
<p>It is not clear why more than one ellipsis should be allowed, but this is the
behavior of higher rank arrays that we are trying to preserve.</p>
</section>
<section id="refactoring">
<h3>Refactoring<a class="headerlink" href="#refactoring" title="Link to this heading">#</a></h3>
<p>Currently all indexing on zero-rank arrays is implemented in a special <code class="docutils literal notranslate"><span class="pre">if</span> <span class="pre">(nd</span>
<span class="pre">==</span> <span class="pre">0)</span></code> branch of code that used to always raise an index error. This ensures
that the changes do not affect any existing usage (except, the usage that
relies on exceptions). On the other hand part of motivation for these changes
was to make behavior of ndarrays more uniform and this should allow to
eliminate <code class="docutils literal notranslate"><span class="pre">if</span> <span class="pre">(nd</span> <span class="pre">==</span> <span class="pre">0)</span></code> checks altogether.</p>
</section>
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