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<div class="chapter" id="Sets">
<div class="header">
<p>
Next: <a href="Polynomial-Manipulations.html" accesskey="n" rel="next">Polynomial Manipulations</a>, Previous: <a href="Statistics.html" accesskey="p" rel="prev">Statistics</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<span id="Sets-1"></span><h2 class="chapter">27 Sets</h2>
<p>Octave has a number of functions for managing sets of data. A set is defined
as a collection of unique elements and is typically represented by a vector of
numbers sorted in ascending order. Any vector or matrix can be converted to a
set by removing duplicates through the use of the <code>unique</code> function.
However, it isn’t necessary to explicitly create a set as all of the functions
which operate on sets will convert their input to a set before proceeding.
</p>
<span id="XREFunique"></span><dl class="def">
<dt id="index-unique"><span class="category">: </span><span><em></em> <strong>unique</strong> <em>(<var>x</var>)</em><a href='#index-unique' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-unique-1"><span class="category">: </span><span><em></em> <strong>unique</strong> <em>(<var>x</var>, "rows")</em><a href='#index-unique-1' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-unique-2"><span class="category">: </span><span><em></em> <strong>unique</strong> <em>(…, "sorted")</em><a href='#index-unique-2' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-unique-3"><span class="category">: </span><span><em></em> <strong>unique</strong> <em>(…, "stable")</em><a href='#index-unique-3' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-unique-4"><span class="category">: </span><span><em>[<var>y</var>, <var>i</var>, <var>j</var>] =</em> <strong>unique</strong> <em>(…)</em><a href='#index-unique-4' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-unique-5"><span class="category">: </span><span><em>[<var>y</var>, <var>i</var>, <var>j</var>] =</em> <strong>unique</strong> <em>(…, "first")</em><a href='#index-unique-5' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-unique-6"><span class="category">: </span><span><em>[<var>y</var>, <var>i</var>, <var>j</var>] =</em> <strong>unique</strong> <em>(…, "last")</em><a href='#index-unique-6' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-unique-7"><span class="category">: </span><span><em>[<var>y</var>, <var>i</var>, <var>j</var>] =</em> <strong>unique</strong> <em>(…, "legacy")</em><a href='#index-unique-7' class='copiable-anchor'> ¶</a></span></dt>
<dd><p>Return the unique elements of <var>x</var>.
</p>
<p>If the input <var>x</var> is a column vector then return a column vector;
Otherwise, return a row vector. <var>x</var> may also be a cell array of
strings.
</p>
<p>If the optional argument <code>"rows"</code> is given then return the unique
rows of <var>x</var>. The input must be a 2-D numeric matrix to use this option.
</p>
<p>The optional argument <code>"sorted"</code>/<code>"stable"</code> controls the order
in which unique values appear in the output. The default is
<code>"sorted"</code> and values in the output are placed in ascending order.
The alternative <code>"stable"</code> preserves the order found in the input
<var>x</var>.
</p>
<p>If requested, return column index vectors <var>i</var> and <var>j</var> such that
<code><var>y</var> = <var>x</var>(<var>i</var>)</code> and <code><var>x</var> = <var>y</var>(<var>j</var>)</code>.
</p>
<p>Additionally, if <var>i</var> is a requested output then one of the flags
<code>"first"</code> or <code>"last"</code> may be given. If <code>"last"</code> is
specified, return the highest possible indices in <var>i</var>, otherwise, if
<code>"first"</code> is specified, return the lowest. The default is
<code>"first"</code>.
</p>
<p>Example 1 : sort order
</p>
<div class="example">
<pre class="example">unique ([3, 1, 1, 2])
⇒ [1, 2, 3]
unique ([3, 1, 1, 2], "stable")
⇒ [3, 1, 2]
</pre></div>
<p>Example 2 : index selection
</p>
<div class="example">
<pre class="example">[~, <var>i</var>] = unique ([3, 1, 1, 2], "first")
⇒ <var>i</var> = [2; 4; 1]
[~, <var>i</var>] = unique ([3, 1, 1, 2], "last")
⇒ <var>i</var> = [3; 4; 1]
</pre></div>
<p>Programming Notes: The input flag <code>"legacy"</code> changes the algorithm
to be compatible with <small>MATLAB</small> releases prior to R2012b. Specifically,
The index ordering flag is changed to <code>"last"</code>, and the shape of the
outputs <var>i</var>, <var>j</var> will follow the shape of the input <var>x</var> rather
than always being column vectors.
</p>
<p>The third output, <var>j</var>, has not been implemented yet when the sort
order is <code>"stable"</code>.
</p>
<p><strong>See also:</strong> <a href="Set-Operations.html#XREFunion">union</a>, <a href="Set-Operations.html#XREFintersect">intersect</a>, <a href="Set-Operations.html#XREFsetdiff">setdiff</a>, <a href="Set-Operations.html#XREFsetxor">setxor</a>, <a href="Set-Operations.html#XREFismember">ismember</a>.
</p></dd></dl>
<span id="XREFuniquetol"></span><dl class="def">
<dt id="index-uniquetol"><span class="category">: </span><span><em><var>c</var> =</em> <strong>uniquetol</strong> <em>(<var>A</var>)</em><a href='#index-uniquetol' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-uniquetol-1"><span class="category">: </span><span><em><var>c</var> =</em> <strong>uniquetol</strong> <em>(<var>A</var>, <var>tol</var>)</em><a href='#index-uniquetol-1' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-uniquetol-2"><span class="category">: </span><span><em><var>c</var> =</em> <strong>uniquetol</strong> <em>(…, <var>property</var>, <var>value</var>)</em><a href='#index-uniquetol-2' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-uniquetol-3"><span class="category">: </span><span><em>[<var>c</var>, <var>ia</var>, <var>ic</var>] =</em> <strong>uniquetol</strong> <em>(…)</em><a href='#index-uniquetol-3' class='copiable-anchor'> ¶</a></span></dt>
<dd><p>Return the unique elements of <var>A</var> within tolerance <var>tol</var>.
</p>
<p>Two values, <var>x</var> and <var>y</var>, are within relative tolerance if
<code>abs (<var>x</var> - <var>y</var>) <= <var>tol</var> * max (abs (<var>A</var>(:)))</code>.
</p>
<p>The input <var>A</var> must be a floating point type (double or single).
</p>
<p>If <var>tol</var> is unspecified, the default tolerance is 1e-12 for double
precision input or 1e-6 for single precision input.
</p>
<p>The function may also be called with the following optional property/value
pairs. Property/value pairs must be passed after other input arguments:
</p>
<dl compact="compact">
<dt><span><code>"ByRows"</code> (default: <code>false</code>)</span></dt>
<dd><p>When true, return the unique rows of <var>A</var>. <var>A</var> must be a 2-D array
to use this option. For rows, the criteria for uniqueness is changed to
<code>all (abs (<var>x</var> - <var>y</var>) <= <var>tol</var>*max (abs (<var>A</var>),[],1))</code>
which compares each column component of a row against a column-specific
tolerance.
</p>
</dd>
<dt><span><code>"DataScale"</code></span></dt>
<dd><p>The tolerance test is changed to
<code>abs (<var>x</var> - <var>y</var>) <= <var>tol</var>*<var>DS</var></code> where <var>DS</var> is a
scalar unless the property <code>"ByRows"</code> is true. In that case, <var>DS</var>
can either be a scalar or a vector with a length equal to the number of
columns in <var>A</var>. Using a value of <code>1.0</code> for <var>DS</var> will change
the tolerance from a relative one to an absolute tolerance. Using a value
of <code>Inf</code> will disable testing.
</p>
</dd>
<dt><span><code>"OutputAllIndices"</code> (default: <code>false</code>)</span></dt>
<dd><p>When true, <var>ia</var> is a cell array (not a vector) that contains the indices
for <em>all</em> elements in <var>A</var> that are within tolerance of a value in
<var>C</var>. That is, each cell in <var>ia</var> corresponds to a single unique
value in <var>C</var>, and the values in each cell correspond to locations in
<var>A</var>.
</p></dd>
</dl>
<p>The output <var>c</var> is a row vector if the input <var>A</var> is a row vector.
For all other cases, a column vector is returned.
</p>
<p>The optional output <var>ia</var> is a column index vector such that
<code><var>c</var> = <var>A</var>(<var>ia</var>)</code>. If the <code>"ByRows"</code> property is
true, the condition is <code><var>c</var> = <var>A</var>(<var>ia</var>, :)</code>. If the
<code>"OutputAllIndices"</code> property is true, then the values
<code><var>A</var>(<var>ia</var>{<var>i</var>})</code> are all within tolerance of the unique
value <code><var>c</var>(<var>i</var>)</code>.
</p>
<p>The optional output <var>ic</var> is a column index vector such that
<code><var>A</var> = <var>c</var>(<var>ic</var>)</code> when <var>A</var> is a vector. When <var>A</var>
is a matrix, <code><var>A</var>(:) = <var>c</var>(<var>ic</var>)</code>. If the <code>"ByRows"</code>
property is true then <code><var>A</var> = <var>c</var>(<var>ic</var>,:)</code>.
</p>
<p>Example: small round-off errors require <code>uniquetol</code>, not <code>unique</code>
</p>
<div class="example">
<pre class="example">x = [1:5];
## Inverse_Function (Function (x)) should return exactly x
y = exp (log (x));
D = unique ([x, y])
⇒ [1.0000 2.0000 3.0000 3.0000 4.0000 5.0000 5.0000]
C = uniquetol ([x, y])
⇒ [1 2 3 4 5]
</pre></div>
<p><strong>See also:</strong> <a href="#XREFunique">unique</a>, <a href="Set-Operations.html#XREFunion">union</a>, <a href="Set-Operations.html#XREFintersect">intersect</a>, <a href="Set-Operations.html#XREFsetdiff">setdiff</a>, <a href="Set-Operations.html#XREFsetxor">setxor</a>, <a href="Set-Operations.html#XREFismember">ismember</a>.
</p></dd></dl>
<ul class="section-toc">
<li><a href="Set-Operations.html" accesskey="1">Set Operations</a></li>
</ul>
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Next: <a href="Polynomial-Manipulations.html">Polynomial Manipulations</a>, Previous: <a href="Statistics.html">Statistics</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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