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<section id="nep-5-generalized-universal-functions">
<span id="nep05"></span><h1>NEP 5 — Generalized universal functions<a class="headerlink" href="#nep-5-generalized-universal-functions" title="Link to this heading">#</a></h1>
<dl class="field-list simple">
<dt class="field-odd">Status<span class="colon">:</span></dt>
<dd class="field-odd"><p>Final</p>
</dd>
</dl>
<p>There is a general need for looping over not only functions on scalars
but also over functions on vectors (or arrays), as explained on
<a class="reference external" href="http://scipy.org/scipy/numpy/wiki/GeneralLoopingFunctions">http://scipy.org/scipy/numpy/wiki/GeneralLoopingFunctions</a>. We propose
to realize this concept by generalizing the universal functions
(ufuncs), and provide a C implementation that adds ~500 lines
to the numpy code base. In current (specialized) ufuncs, the elementary
function is limited to element-by-element operations, whereas the
generalized version supports “sub-array” by “sub-array” operations.
The Perl vector library PDL provides a similar functionality and its
terms are re-used in the following.</p>
<p>Each generalized ufunc has information associated with it that states
what the “core” dimensionality of the inputs is, as well as the
corresponding dimensionality of the outputs (the element-wise ufuncs
have zero core dimensions). The list of the core dimensions for all
arguments is called the “signature” of a ufunc. For example, the
ufunc numpy.add has signature <code class="docutils literal notranslate"><span class="pre">(),()->()</span></code> defining two scalar inputs
and one scalar output.</p>
<p>Another example is (see the GeneralLoopingFunctions page) the function
<code class="docutils literal notranslate"><span class="pre">inner1d(a,b)</span></code> with a signature of <code class="docutils literal notranslate"><span class="pre">(i),(i)->()</span></code>. This applies the
inner product along the last axis of each input, but keeps the
remaining indices intact. For example, where <code class="docutils literal notranslate"><span class="pre">a</span></code> is of shape <code class="docutils literal notranslate"><span class="pre">(3,5,N)</span></code>
and <code class="docutils literal notranslate"><span class="pre">b</span></code> is of shape <code class="docutils literal notranslate"><span class="pre">(5,N)</span></code>, this will return an output of shape <code class="docutils literal notranslate"><span class="pre">(3,5)</span></code>.
The underlying elementary function is called 3*5 times. In the
signature, we specify one core dimension <code class="docutils literal notranslate"><span class="pre">(i)</span></code> for each input and zero core
dimensions <code class="docutils literal notranslate"><span class="pre">()</span></code> for the output, since it takes two 1-d arrays and
returns a scalar. By using the same name <code class="docutils literal notranslate"><span class="pre">i</span></code>, we specify that the two
corresponding dimensions should be of the same size (or one of them is
of size 1 and will be broadcasted).</p>
<p>The dimensions beyond the core dimensions are called “loop” dimensions. In
the above example, this corresponds to <code class="docutils literal notranslate"><span class="pre">(3,5)</span></code>.</p>
<p>The usual numpy “broadcasting” rules apply, where the signature
determines how the dimensions of each input/output object are split
into core and loop dimensions:</p>
<ol class="arabic simple">
<li><p>While an input array has a smaller dimensionality than the corresponding
number of core dimensions, 1’s are prepended to its shape.</p></li>
<li><p>The core dimensions are removed from all inputs and the remaining
dimensions are broadcasted; defining the loop dimensions.</p></li>
<li><p>The output is given by the loop dimensions plus the output core dimensions.</p></li>
</ol>
<section id="definitions">
<h2>Definitions<a class="headerlink" href="#definitions" title="Link to this heading">#</a></h2>
<dl class="simple">
<dt>Elementary Function</dt><dd><p>Each ufunc consists of an elementary function that performs the
most basic operation on the smallest portion of array arguments
(e.g. adding two numbers is the most basic operation in adding two
arrays). The ufunc applies the elementary function multiple times
on different parts of the arrays. The input/output of elementary
functions can be vectors; e.g., the elementary function of inner1d
takes two vectors as input.</p>
</dd>
<dt>Signature</dt><dd><p>A signature is a string describing the input/output dimensions of
the elementary function of a ufunc. See section below for more
details.</p>
</dd>
<dt>Core Dimension</dt><dd><p>The dimensionality of each input/output of an elementary function
is defined by its core dimensions (zero core dimensions correspond
to a scalar input/output). The core dimensions are mapped to the
last dimensions of the input/output arrays.</p>
</dd>
<dt>Dimension Name</dt><dd><p>A dimension name represents a core dimension in the signature.
Different dimensions may share a name, indicating that they are of
the same size (or are broadcastable).</p>
</dd>
<dt>Dimension Index</dt><dd><p>A dimension index is an integer representing a dimension name. It
enumerates the dimension names according to the order of the first
occurrence of each name in the signature.</p>
</dd>
</dl>
</section>
<section id="details-of-signature">
<h2>Details of signature<a class="headerlink" href="#details-of-signature" title="Link to this heading">#</a></h2>
<p>The signature defines “core” dimensionality of input and output
variables, and thereby also defines the contraction of the
dimensions. The signature is represented by a string of the
following format:</p>
<ul class="simple">
<li><p>Core dimensions of each input or output array are represented by a
list of dimension names in parentheses, <code class="docutils literal notranslate"><span class="pre">(i_1,...,i_N)</span></code>; a scalar
input/output is denoted by <code class="docutils literal notranslate"><span class="pre">()</span></code>. Instead of <code class="docutils literal notranslate"><span class="pre">i_1</span></code>, <code class="docutils literal notranslate"><span class="pre">i_2</span></code>,
etc, one can use any valid Python variable name.</p></li>
<li><p>Dimension lists for different arguments are separated by <code class="docutils literal notranslate"><span class="pre">","</span></code>.
Input/output arguments are separated by <code class="docutils literal notranslate"><span class="pre">"->"</span></code>.</p></li>
<li><p>If one uses the same dimension name in multiple locations, this
enforces the same size (or broadcastable size) of the corresponding
dimensions.</p></li>
</ul>
<p>The formal syntax of signatures is as follows:</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">Dimension</span> <span class="n">name</span><span class="o">></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="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">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>
</pre></div>
</div>
<p>Notes:</p>
<ol class="arabic simple">
<li><p>All quotes are for clarity.</p></li>
<li><p>Core dimensions that share the same name must be broadcastable, as
the two <code class="docutils literal notranslate"><span class="pre">i</span></code> in our example above. Each dimension name typically
corresponding to one level of looping in the elementary function’s
implementation.</p></li>
<li><p>White spaces are ignored.</p></li>
</ol>
<p>Here are some examples of signatures:</p>
<div class="pst-scrollable-table-container"><table class="table">
<tbody>
<tr class="row-odd"><td><p>add</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">(),()->()</span></code></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>inner1d</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">(i),(i)->()</span></code></p></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>sum1d</p></td>
<td><p><code class="docutils literal notranslate"><span class="pre">(i)->()</span></code></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>dot2d</p></td>
<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>outer_inner</p></td>
<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="c-api-for-implementing-elementary-functions">
<h2>C-API for implementing elementary functions<a class="headerlink" href="#c-api-for-implementing-elementary-functions" title="Link to this heading">#</a></h2>
<p>The current interface remains unchanged, and <code class="docutils literal notranslate"><span class="pre">PyUFunc_FromFuncAndData</span></code>
can still be used to implement (specialized) ufuncs, consisting of
scalar elementary functions.</p>
<p>One can use <code class="docutils literal notranslate"><span class="pre">PyUFunc_FromFuncAndDataAndSignature</span></code> to declare a more
general ufunc. The argument list is the same as
<code class="docutils literal notranslate"><span class="pre">PyUFunc_FromFuncAndData</span></code>, with an additional argument specifying the
signature as C string.</p>
<p>Furthermore, the callback function is of the same type as before,
<code class="docutils literal notranslate"><span class="pre">void</span> <span class="pre">(*foo)(char</span> <span class="pre">**args,</span> <span class="pre">intp</span> <span class="pre">*dimensions,</span> <span class="pre">intp</span> <span class="pre">*steps,</span> <span class="pre">void</span> <span class="pre">*func)</span></code>.
When invoked, <code class="docutils literal notranslate"><span class="pre">args</span></code> is a list of length <code class="docutils literal notranslate"><span class="pre">nargs</span></code> containing
the data of all input/output arguments. For a scalar elementary
function, <code class="docutils literal notranslate"><span class="pre">steps</span></code> is also of length <code class="docutils literal notranslate"><span class="pre">nargs</span></code>, denoting the strides used
for the arguments. <code class="docutils literal notranslate"><span class="pre">dimensions</span></code> is a pointer to a single integer
defining the size of the axis to be looped over.</p>
<p>For a non-trivial signature, <code class="docutils literal notranslate"><span class="pre">dimensions</span></code> will also contain the sizes
of the core dimensions as well, starting at the second entry. Only
one size is provided for each unique dimension name and the sizes are
given according to the first occurrence of a dimension name in the
signature.</p>
<p>The first <code class="docutils literal notranslate"><span class="pre">nargs</span></code> elements of <code class="docutils literal notranslate"><span class="pre">steps</span></code> remain the same as for scalar
ufuncs. The following elements contain the strides of all core
dimensions for all arguments in order.</p>
<p>For example, consider a ufunc with signature <code class="docutils literal notranslate"><span class="pre">(i,j),(i)->()</span></code>. In
this case, <code class="docutils literal notranslate"><span class="pre">args</span></code> will contain three pointers to the data of the
input/output arrays <code class="docutils literal notranslate"><span class="pre">a</span></code>, <code class="docutils literal notranslate"><span class="pre">b</span></code>, <code class="docutils literal notranslate"><span class="pre">c</span></code>. Furthermore, <code class="docutils literal notranslate"><span class="pre">dimensions</span></code> will be
<code class="docutils literal notranslate"><span class="pre">[N,</span> <span class="pre">I,</span> <span class="pre">J]</span></code> to define the size of <code class="docutils literal notranslate"><span class="pre">N</span></code> of the loop and the sizes <code class="docutils literal notranslate"><span class="pre">I</span></code> and <code class="docutils literal notranslate"><span class="pre">J</span></code>
for the core dimensions <code class="docutils literal notranslate"><span class="pre">i</span></code> and <code class="docutils literal notranslate"><span class="pre">j</span></code>. Finally, <code class="docutils literal notranslate"><span class="pre">steps</span></code> will be
<code class="docutils literal notranslate"><span class="pre">[a_N,</span> <span class="pre">b_N,</span> <span class="pre">c_N,</span> <span class="pre">a_i,</span> <span class="pre">a_j,</span> <span class="pre">b_i]</span></code>, containing all necessary strides.</p>
</section>
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