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<h3 class="section" id="Special-Utility-Matrices-1"><span>16.3 Special Utility Matrices<a class="copiable-link" href="#Special-Utility-Matrices-1"> ¶</a></span></h3>
<a class="anchor" id="XREFeye"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-eye"><span><code class="def-type"><var class="var">I</var> =</code> <strong class="def-name">eye</strong> <code class="def-code-arguments">(<var class="var">n</var>)</code><a class="copiable-link" href="#index-eye"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-eye-1"><span><code class="def-type"><var class="var">I</var> =</code> <strong class="def-name">eye</strong> <code class="def-code-arguments">(<var class="var">m</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-eye-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-eye-2"><span><code class="def-type"><var class="var">I</var> =</code> <strong class="def-name">eye</strong> <code class="def-code-arguments">([<var class="var">m</var> <var class="var">n</var>])</code><a class="copiable-link" href="#index-eye-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-eye-3"><span><code class="def-type"><var class="var">I</var> =</code> <strong class="def-name">eye</strong> <code class="def-code-arguments">(…, <var class="var">class</var>)</code><a class="copiable-link" href="#index-eye-3"> ¶</a></span></dt>
<dd><p>Return an identity matrix.
</p>
<p>If invoked with a single scalar argument <var class="var">n</var>, return a square
NxN identity matrix.
</p>
<p>If supplied two scalar arguments (<var class="var">m</var>, <var class="var">n</var>), <code class="code">eye</code> takes them
to be the number of rows and columns. If given a vector with two elements,
<code class="code">eye</code> uses the values of the elements as the number of rows and
columns, respectively. For example:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">eye (3)
⇒ 1 0 0
0 1 0
0 0 1
</pre></div></div>
<p>The following expressions all produce the same result:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">eye (2)
≡
eye (2, 2)
≡
eye (size ([1, 2; 3, 4]))
</pre></div></div>
<p>The optional argument <var class="var">class</var>, allows <code class="code">eye</code> to return an array of
the specified type, like
</p>
<div class="example">
<pre class="example-preformatted">val = zeros (n,m, "uint8")
</pre></div>
<p>Calling <code class="code">eye</code> with no arguments is equivalent to calling it with an
argument of 1. Any negative dimensions are treated as zero. These odd
definitions are for compatibility with <small class="sc">MATLAB</small>.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="Creating-Sparse-Matrices.html#XREFspeye">speye</a>, <a class="ref" href="#XREFones">ones</a>, <a class="ref" href="#XREFzeros">zeros</a>.
</p></dd></dl>
<a class="anchor" id="XREFones"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-ones"><span><code class="def-type"><var class="var">val</var> =</code> <strong class="def-name">ones</strong> <code class="def-code-arguments">(<var class="var">n</var>)</code><a class="copiable-link" href="#index-ones"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-ones-1"><span><code class="def-type"><var class="var">val</var> =</code> <strong class="def-name">ones</strong> <code class="def-code-arguments">(<var class="var">m</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-ones-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-ones-2"><span><code class="def-type"><var class="var">val</var> =</code> <strong class="def-name">ones</strong> <code class="def-code-arguments">(<var class="var">m</var>, <var class="var">n</var>, <var class="var">k</var>, …)</code><a class="copiable-link" href="#index-ones-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-ones-3"><span><code class="def-type"><var class="var">val</var> =</code> <strong class="def-name">ones</strong> <code class="def-code-arguments">([<var class="var">m</var> <var class="var">n</var> …])</code><a class="copiable-link" href="#index-ones-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-ones-4"><span><code class="def-type"><var class="var">val</var> =</code> <strong class="def-name">ones</strong> <code class="def-code-arguments">(…, "like", <var class="var">var</var>)</code><a class="copiable-link" href="#index-ones-4"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-ones-5"><span><code class="def-type"><var class="var">val</var> =</code> <strong class="def-name">ones</strong> <code class="def-code-arguments">(…, <var class="var">class</var>)</code><a class="copiable-link" href="#index-ones-5"> ¶</a></span></dt>
<dd><p>Return a matrix or N-dimensional array whose elements are all 1.
</p>
<p>If invoked with a single scalar integer argument <var class="var">n</var>, return a square
NxN matrix.
</p>
<p>If invoked with two or more scalar integer arguments, or a vector of integer
values, return an array with the given dimensions.
</p>
<p>To create a constant matrix whose values are all the same use an expression
such as
</p>
<div class="example">
<pre class="example-preformatted">val_matrix = val * ones (m, n)
</pre></div>
<p>If a variable <var class="var">var</var> is specified after <code class="code">"like"</code>, the output <var class="var">val</var>
will have the same data type, complexity, and sparsity as <var class="var">var</var>.
</p>
<p>The optional argument <var class="var">class</var> specifies the class of the return array
and defaults to double. For example:
</p>
<div class="example">
<pre class="example-preformatted">val = ones (m,n, "uint8")
</pre></div>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFzeros">zeros</a>.
</p></dd></dl>
<a class="anchor" id="XREFzeros"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-zeros"><span><code class="def-type"><var class="var">val</var> =</code> <strong class="def-name">zeros</strong> <code class="def-code-arguments">(<var class="var">n</var>)</code><a class="copiable-link" href="#index-zeros"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-zeros-1"><span><code class="def-type"><var class="var">val</var> =</code> <strong class="def-name">zeros</strong> <code class="def-code-arguments">(<var class="var">m</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-zeros-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-zeros-2"><span><code class="def-type"><var class="var">val</var> =</code> <strong class="def-name">zeros</strong> <code class="def-code-arguments">(<var class="var">m</var>, <var class="var">n</var>, <var class="var">k</var>, …)</code><a class="copiable-link" href="#index-zeros-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-zeros-3"><span><code class="def-type"><var class="var">val</var> =</code> <strong class="def-name">zeros</strong> <code class="def-code-arguments">([<var class="var">m</var> <var class="var">n</var> …])</code><a class="copiable-link" href="#index-zeros-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-zeros-4"><span><code class="def-type"><var class="var">val</var> =</code> <strong class="def-name">zeros</strong> <code class="def-code-arguments">(…, "like", <var class="var">var</var>)</code><a class="copiable-link" href="#index-zeros-4"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-zeros-5"><span><code class="def-type"><var class="var">val</var> =</code> <strong class="def-name">zeros</strong> <code class="def-code-arguments">(…, <var class="var">class</var>)</code><a class="copiable-link" href="#index-zeros-5"> ¶</a></span></dt>
<dd><p>Return a matrix or N-dimensional array whose elements are all 0.
</p>
<p>If invoked with a single scalar integer argument, return a square
NxN matrix.
</p>
<p>If invoked with two or more scalar integer arguments, or a vector of integer
values, return an array with the given dimensions.
</p>
<p>If a variable <var class="var">var</var> is specified after <code class="code">"like"</code>, the output <var class="var">val</var>
will have the same data type, complexity, and sparsity as <var class="var">var</var>.
</p>
<p>The optional argument <var class="var">class</var> specifies the class of the return array
and defaults to double. For example:
</p>
<div class="example">
<pre class="example-preformatted">val = zeros (m,n, "uint8")
</pre></div>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFones">ones</a>.
</p></dd></dl>
<a class="anchor" id="XREFrepmat"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-repmat"><span><code class="def-type"><var class="var">B</var> =</code> <strong class="def-name">repmat</strong> <code class="def-code-arguments">(<var class="var">A</var>, <var class="var">m</var>)</code><a class="copiable-link" href="#index-repmat"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-repmat-1"><span><code class="def-type"><var class="var">B</var> =</code> <strong class="def-name">repmat</strong> <code class="def-code-arguments">(<var class="var">A</var>, <var class="var">m</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-repmat-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-repmat-2"><span><code class="def-type"><var class="var">B</var> =</code> <strong class="def-name">repmat</strong> <code class="def-code-arguments">(<var class="var">A</var>, <var class="var">m</var>, <var class="var">n</var>, <var class="var">p</var> …)</code><a class="copiable-link" href="#index-repmat-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-repmat-3"><span><code class="def-type"><var class="var">B</var> =</code> <strong class="def-name">repmat</strong> <code class="def-code-arguments">(<var class="var">A</var>, [<var class="var">m</var> <var class="var">n</var>])</code><a class="copiable-link" href="#index-repmat-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-repmat-4"><span><code class="def-type"><var class="var">B</var> =</code> <strong class="def-name">repmat</strong> <code class="def-code-arguments">(<var class="var">A</var>, [<var class="var">m</var> <var class="var">n</var> <var class="var">p</var> …])</code><a class="copiable-link" href="#index-repmat-4"> ¶</a></span></dt>
<dd><p>Repeat matrix or N-D array.
</p>
<p>Form a block matrix of size <var class="var">m</var> by <var class="var">n</var>, with a copy of matrix
<var class="var">A</var> as each element.
</p>
<p>If <var class="var">n</var> is not specified, form an <var class="var">m</var> by <var class="var">m</var> block matrix. For
copying along more than two dimensions, specify the number of times to copy
across each dimension <var class="var">m</var>, <var class="var">n</var>, <var class="var">p</var>, …, in a vector in the
second argument.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="Broadcasting.html#XREFbsxfun">bsxfun</a>, <a class="ref" href="Functions-of-a-Matrix.html#XREFkron">kron</a>, <a class="ref" href="#XREFrepelems">repelems</a>.
</p></dd></dl>
<a class="anchor" id="XREFrepelems"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-repelems"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">repelems</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">r</var>)</code><a class="copiable-link" href="#index-repelems"> ¶</a></span></dt>
<dd><p>Construct a vector of repeated elements from <var class="var">x</var>.
</p>
<p><var class="var">r</var> is a 2x<var class="var">N</var> integer matrix specifying which elements to repeat
and how often to repeat each element. Entries in the first row,
<var class="var">r</var>(1,j), select an element to repeat. The corresponding entry in the
second row, <var class="var">r</var>(2,j), specifies the repeat count. If <var class="var">x</var> is a
matrix then the columns of <var class="var">x</var> are imagined to be stacked on top of
each other for purposes of the selection index. A row vector is always
returned.
</p>
<p>Conceptually the result is calculated as follows:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">y = [];
for i = 1:columns (<var class="var">r</var>)
y = [y, <var class="var">x</var>(<var class="var">r</var>(1,i)*ones(1, <var class="var">r</var>(2,i)))];
endfor
</pre></div></div>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFrepmat">repmat</a>, <a class="ref" href="Rearranging-Matrices.html#XREFcat">cat</a>.
</p></dd></dl>
<a class="anchor" id="XREFrepelem"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-repelem"><span><code class="def-type"><var class="var">xxx</var> =</code> <strong class="def-name">repelem</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">R</var>)</code><a class="copiable-link" href="#index-repelem"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-repelem-1"><span><code class="def-type"><var class="var">xxx</var> =</code> <strong class="def-name">repelem</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">R_1</var>, …, <var class="var">R_n</var>)</code><a class="copiable-link" href="#index-repelem-1"> ¶</a></span></dt>
<dd><p>Construct an array of repeated elements from <var class="var">x</var> and repeat instructions
<var class="var">R_1</var>, <small class="enddots">...</small>
</p>
<p><var class="var">x</var> must be a scalar, vector, or N-dimensional array.
</p>
<p>A repeat instruction <var class="var">R_j</var> must either be a scalar or a vector. If the
instruction is a scalar then each component of <var class="var">x</var> in dimension <var class="var">j</var>
is repeated <var class="var">R_j</var> times. If the instruction is a vector then it must
have the same number of elements as the corresponding dimension <var class="var">j</var> of
<var class="var">x</var>. In this case, the <var class="var">k</var>th component of dimension <var class="var">j</var> is
repeated <code class="code"><var class="var">R_j</var>(<var class="var">k</var>)</code> times.
</p>
<p>If <var class="var">x</var> is a scalar or vector then <code class="code">repelem</code> may be called with just
a single repeat instruction <var class="var">R</var> and <code class="code">repelem</code> will return a vector
with the same orientation as the input.
</p>
<p>If <var class="var">x</var> is a matrix then at least two <var class="var">R_j</var>s must be specified.
</p>
<p>Note: Using <code class="code">repelem</code> with a vector <var class="var">x</var> and a vector for <var class="var">R_j</var>
is equivalent to Run Length Decoding.
</p>
<p>Examples:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">A = [1 2 3 4 5];
B = [2 1 0 1 2];
repelem (A, B)
⇒ 1 1 2 4 5 5
</pre></div></div>
<div class="example">
<div class="group"><pre class="example-preformatted">A = magic (3)
⇒ A =
8 1 6
3 5 7
4 9 2
B1 = [1 2 3];
B2 = 2;
repelem (A, B1, B2)
⇒ 8 8 1 1 6 6
3 3 5 5 7 7
3 3 5 5 7 7
4 4 9 9 2 2
4 4 9 9 2 2
4 4 9 9 2 2
</pre></div></div>
<p>More <var class="var">R_j</var> may be specified than the number of dimensions of <var class="var">x</var>.
Any excess <var class="var">R_j</var> must be scalars (because <var class="var">x</var>’s size in those
dimensions is only 1), and <var class="var">x</var> will be replicated in those dimensions
accordingly.
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">A = [1 2 3 4 5];
B1 = 2;
B2 = [2 1 3 0 2];
B3 = 3;
repelem (A, B1, B2, B3)
⇒ ans(:,:,1) =
1 1 2 3 3 3 5 5
1 1 2 3 3 3 5 5
ans(:,:,2) =
1 1 2 3 3 3 5 5
1 1 2 3 3 3 5 5
ans(:,:,3) =
1 1 2 3 3 3 5 5
1 1 2 3 3 3 5 5
</pre></div></div>
<p><var class="var">R_j</var> must be specified in order. A placeholder of 1 may be used for
dimensions which do not need replication.
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">repelem ([-1, 0; 0, 1], 1, 2, 1, 2)
⇒ ans(:,:,1,1) =
-1 -1 0 0
0 0 1 1
ans(:,:,1,2) =
-1 -1 0 0
0 0 1 1
</pre></div></div>
<p>If fewer <var class="var">R_j</var> are given than the number of dimensions in <var class="var">x</var>,
<code class="code">repelem</code> will assume <var class="var">R_j</var> is 1 for those dimensions.
</p>
<div class="example">
<pre class="example-preformatted">A = cat (3, [-1 0; 0 1], [-1 0; 0 1])
⇒ ans(:,:,1) =
-1 0
0 1
ans(:,:,2) =
-1 0
0 1
repelem (A,2,3)
⇒ ans(:,:,1) =
-1 -1 -1 0 0 0
-1 -1 -1 0 0 0
0 0 0 1 1 1
0 0 0 1 1 1
ans(:,:,2) =
-1 -1 -1 0 0 0
-1 -1 -1 0 0 0
0 0 0 1 1 1
0 0 0 1 1 1
</pre></div>
<p><code class="code">repelem</code> preserves the class of <var class="var">x</var>, and works with strings,
cell arrays, NA, and NAN inputs. If any <var class="var">R_j</var> is 0 the output will
be an empty array.
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">repelem ("Octave", 2, 3)
⇒ OOOccctttaaavvveee
OOOccctttaaavvveee
repelem ([1 2 3; 1 2 3], 2, 0)
⇒ [](4x0)
</pre></div></div>
<p><strong class="strong">See also:</strong> <a class="ref" href="Rearranging-Matrices.html#XREFcat">cat</a>, <a class="ref" href="Functions-of-a-Matrix.html#XREFkron">kron</a>, <a class="ref" href="#XREFrepmat">repmat</a>.
</p></dd></dl>
<p>The functions <code class="code">linspace</code> and <code class="code">logspace</code> make it very easy to
create vectors with evenly or logarithmically spaced elements.
See <a class="xref" href="Ranges.html">Ranges</a>.
</p>
<a class="anchor" id="XREFlinspace"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-linspace"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">linspace</strong> <code class="def-code-arguments">(<var class="var">start</var>, <var class="var">end</var>)</code><a class="copiable-link" href="#index-linspace"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-linspace-1"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">linspace</strong> <code class="def-code-arguments">(<var class="var">start</var>, <var class="var">end</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-linspace-1"> ¶</a></span></dt>
<dd><p>Return a row vector with <var class="var">n</var> linearly spaced elements between <var class="var">start</var>
and <var class="var">end</var>.
</p>
<p>If the number of elements <var class="var">n</var> is greater than one, then the endpoints
<var class="var">start</var> and <var class="var">end</var> are always included in the range. If <var class="var">start</var> is
greater than <var class="var">end</var>, the elements are stored in decreasing order. If the
number of points <var class="var">n</var> is not specified, a value of 100 is used.
</p>
<p>The <code class="code">linspace</code> function returns a row vector when both <var class="var">start</var> and
<var class="var">end</var> are scalars. If one, or both, inputs are vectors, then
<code class="code">linspace</code> transforms them to column vectors and returns a matrix where
each row is an independent sequence between
<code class="code"><var class="var">start</var>(<var class="var">row_n</var>), <var class="var">end</var>(<var class="var">row_n</var>)</code><!-- /@w -->.
</p>
<p>Programming Notes: For compatibility with <small class="sc">MATLAB</small>, return the second
argument (<var class="var">end</var>) when a single value (<var class="var">n</var> = 1) is requested. If
<var class="var">n</var> is not an integer then <code class="code">floor (<var class="var">n</var>)</code> is used to round the
number of elements. If <var class="var">n</var> is zero or negative then an empty 1x0 matrix
is returned.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="Defining-Indexing-And-Indexed-Assignment.html#XREFcolon">colon</a>, <a class="ref" href="#XREFlogspace">logspace</a>.
</p></dd></dl>
<a class="anchor" id="XREFlogspace"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-logspace"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">logspace</strong> <code class="def-code-arguments">(<var class="var">a</var>, <var class="var">b</var>)</code><a class="copiable-link" href="#index-logspace"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-logspace-1"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">logspace</strong> <code class="def-code-arguments">(<var class="var">a</var>, <var class="var">b</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-logspace-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-logspace-2"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">logspace</strong> <code class="def-code-arguments">(<var class="var">a</var>, pi)</code><a class="copiable-link" href="#index-logspace-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-logspace-3"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">logspace</strong> <code class="def-code-arguments">(<var class="var">a</var>, pi, <var class="var">n</var>)</code><a class="copiable-link" href="#index-logspace-3"> ¶</a></span></dt>
<dd><p>Return a row vector with <var class="var">n</var> elements logarithmically spaced from
10^<var class="var">a</var> to 10^<var class="var">b</var>.
</p>
<p>If the number of elements <var class="var">n</var> is unspecified it defaults to 50.
</p>
<p>If <var class="var">b</var> is equal to
pi,
the points are between
10^<var class="var">a</var> and pi,
<em class="emph">not</em>
10^<var class="var">a</var> and 10^pi,
which is useful in digital signal processing.
</p>
<p>Programming Notes: For compatibility with <small class="sc">MATLAB</small>, return the right-hand
side of the range
(10^<var class="var">b</var>)
when a single value (<var class="var">n</var> = 1) is requested.
If <var class="var">n</var> is not an integer then <code class="code">floor (<var class="var">n</var>)</code> is used to round
the number of elements. If <var class="var">n</var> is zero or negative then an empty 1x0
matrix is returned.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFlinspace">linspace</a>.
</p></dd></dl>
<a class="anchor" id="XREFrand"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-rand"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">rand</strong> <code class="def-code-arguments">(<var class="var">n</var>)</code><a class="copiable-link" href="#index-rand"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rand-1"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">rand</strong> <code class="def-code-arguments">(<var class="var">m</var>, <var class="var">n</var>, …)</code><a class="copiable-link" href="#index-rand-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rand-2"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">rand</strong> <code class="def-code-arguments">([<var class="var">m</var> <var class="var">n</var> …])</code><a class="copiable-link" href="#index-rand-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rand-3"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">rand</strong> <code class="def-code-arguments">(…, "single")</code><a class="copiable-link" href="#index-rand-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rand-4"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">rand</strong> <code class="def-code-arguments">(…, "double")</code><a class="copiable-link" href="#index-rand-4"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rand-5"><span><code class="def-type"><var class="var">v</var> =</code> <strong class="def-name">rand</strong> <code class="def-code-arguments">("state")</code><a class="copiable-link" href="#index-rand-5"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rand-6"><span><strong class="def-name">rand</strong> <code class="def-code-arguments">("state", <var class="var">v</var>)</code><a class="copiable-link" href="#index-rand-6"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rand-7"><span><strong class="def-name">rand</strong> <code class="def-code-arguments">("state", "reset")</code><a class="copiable-link" href="#index-rand-7"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rand-8"><span><code class="def-type"><var class="var">v</var> =</code> <strong class="def-name">rand</strong> <code class="def-code-arguments">("seed")</code><a class="copiable-link" href="#index-rand-8"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rand-9"><span><strong class="def-name">rand</strong> <code class="def-code-arguments">("seed", <var class="var">v</var>)</code><a class="copiable-link" href="#index-rand-9"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rand-10"><span><strong class="def-name">rand</strong> <code class="def-code-arguments">("seed", "reset")</code><a class="copiable-link" href="#index-rand-10"> ¶</a></span></dt>
<dd><p>Return a matrix with random elements uniformly distributed on the
interval (0, 1).
</p>
<p>The arguments are handled the same as the arguments for <code class="code">eye</code>.
</p>
<p>You can query the state of the random number generator using the form
</p>
<div class="example">
<pre class="example-preformatted">v = rand ("state")
</pre></div>
<p>This returns a column vector <var class="var">v</var> of length 625. Later, you can restore
the random number generator to the state <var class="var">v</var> using the form
</p>
<div class="example">
<pre class="example-preformatted">rand ("state", v)
</pre></div>
<p>You may also initialize the state vector from an arbitrary vector of length
≤ 625 for <var class="var">v</var>. This new state will be a hash based on the value of
<var class="var">v</var>, not <var class="var">v</var> itself.
</p>
<p>By default, the generator is initialized by contributing entropy from the
wall clock time, the CPU time, the current fraction of a second, the process
ID and—if available—up to 1024 bits from the C++ random numbers source
<code class="code">random_device</code>, which might be non-deterministic (implementation
specific). Note that this differs from <small class="sc">MATLAB</small>, which always initializes
the state to the same state at startup. To obtain behavior comparable to
<small class="sc">MATLAB</small>, initialize with a deterministic state vector in Octave’s startup
files (see <a class="pxref" href="Startup-Files.html">Startup Files</a>).
</p>
<p>To compute the pseudo-random sequence, <code class="code">rand</code> uses the Mersenne
Twister with a period of <em class="math">2^{19937}-1</em>
(See M. Matsumoto and T. Nishimura,
<cite class="cite">Mersenne Twister: A 623-dimensionally equidistributed uniform
pseudorandom number generator</cite>,
ACM Trans. on Modeling and Computer Simulation Vol. 8, No. 1,
pp. 3–30, January 1998,
<a class="url" href="http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html">http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html</a>).
Do <strong class="strong">not</strong> use for cryptography without securely hashing several
returned values together, otherwise the generator state can be learned after
reading 624 consecutive values.
</p>
<p>Older versions of Octave used a different random number generator.
The new generator is used by default as it is significantly faster than the
old generator, and produces random numbers with a significantly longer cycle
time. However, in some circumstances it might be desirable to obtain the
same random sequences as produced by the old generators. To do this the
keyword <code class="code">"seed"</code> is used to specify that the old generators should
be used, as in
</p>
<div class="example">
<pre class="example-preformatted">rand ("seed", val)
</pre></div>
<p>which sets the seed of the generator to <var class="var">val</var>. The seed of the
generator can be queried with
</p>
<div class="example">
<pre class="example-preformatted">s = rand ("seed")
</pre></div>
<p>However, it should be noted that querying the seed will not cause
<code class="code">rand</code> to use the old generators, only setting the seed will. To cause
<code class="code">rand</code> to once again use the new generators, the keyword
<code class="code">"state"</code> should be used to reset the state of the <code class="code">rand</code>.
</p>
<p>The state or seed of the generator can be reset to a new random value using
the <code class="code">"reset"</code> keyword.
</p>
<p>The class of the value returned can be controlled by a trailing
<code class="code">"double"</code> or <code class="code">"single"</code> argument. These are the only valid
classes.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFrandn">randn</a>, <a class="ref" href="#XREFrande">rande</a>, <a class="ref" href="#XREFrandg">randg</a>, <a class="ref" href="#XREFrandp">randp</a>.
</p></dd></dl>
<a class="anchor" id="XREFrandi"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-randi"><span><code class="def-type"><var class="var">R</var> =</code> <strong class="def-name">randi</strong> <code class="def-code-arguments">(<var class="var">imax</var>)</code><a class="copiable-link" href="#index-randi"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randi-1"><span><code class="def-type"><var class="var">R</var> =</code> <strong class="def-name">randi</strong> <code class="def-code-arguments">(<var class="var">imax</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-randi-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randi-2"><span><code class="def-type"><var class="var">R</var> =</code> <strong class="def-name">randi</strong> <code class="def-code-arguments">(<var class="var">imax</var>, <var class="var">m</var>, <var class="var">n</var>, …)</code><a class="copiable-link" href="#index-randi-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randi-3"><span><code class="def-type"><var class="var">R</var> =</code> <strong class="def-name">randi</strong> <code class="def-code-arguments">([<var class="var">imin</var> <var class="var">imax</var>], …)</code><a class="copiable-link" href="#index-randi-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randi-4"><span><code class="def-type"><var class="var">R</var> =</code> <strong class="def-name">randi</strong> <code class="def-code-arguments">(…, "<var class="var">class</var>")</code><a class="copiable-link" href="#index-randi-4"> ¶</a></span></dt>
<dd><p>Return random integers in the range 1:<var class="var">imax</var>.
</p>
<p>Additional arguments determine the shape of the return matrix. When no
arguments are specified a single random integer is returned. If one
argument <var class="var">n</var> is specified then a square matrix (<var class="var">n</var> x <var class="var">n</var>)<!-- /@w -->
is returned. Two or more arguments will return a multi-dimensional matrix
(<var class="var">m</var> x <var class="var">n</var> x …)<!-- /@w -->.
</p>
<p>The integer range may optionally be described by a two-element matrix with a
lower and upper bound in which case the returned integers will be on the
interval [<var class="var">imin</var>, <var class="var">imax</var>]<!-- /@w -->.
</p>
<p>The optional argument <var class="var">class</var> will return a matrix of the requested
type. The default is <code class="code">"double"</code>.
</p>
<p>The following example returns 150 integers in the range 1–10.
</p>
<div class="example">
<pre class="example-preformatted">ri = randi (10, 150, 1)
</pre></div>
<p>Implementation Note: <code class="code">randi</code> relies internally on <code class="code">rand</code> which
uses class <code class="code">"double"</code> to represent numbers. This limits the maximum
integer (<var class="var">imax</var>) and range (<var class="var">imax</var> - <var class="var">imin</var>) to the value
returned by the <code class="code">flintmax</code> function. For IEEE 754 floating point
numbers this value is <em class="math">2^{53} - 1</em><!-- /@w -->.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFrand">rand</a>, <a class="ref" href="#XREFrandn">randn</a>.
</p></dd></dl>
<a class="anchor" id="XREFrandn"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-randn"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">randn</strong> <code class="def-code-arguments">(<var class="var">n</var>)</code><a class="copiable-link" href="#index-randn"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randn-1"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">randn</strong> <code class="def-code-arguments">(<var class="var">m</var>, <var class="var">n</var>, …)</code><a class="copiable-link" href="#index-randn-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randn-2"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">randn</strong> <code class="def-code-arguments">([<var class="var">m</var> <var class="var">n</var> …])</code><a class="copiable-link" href="#index-randn-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randn-3"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">randn</strong> <code class="def-code-arguments">(…, "single")</code><a class="copiable-link" href="#index-randn-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randn-4"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">randn</strong> <code class="def-code-arguments">(…, "double")</code><a class="copiable-link" href="#index-randn-4"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randn-5"><span><code class="def-type"><var class="var">v</var> =</code> <strong class="def-name">randn</strong> <code class="def-code-arguments">("state")</code><a class="copiable-link" href="#index-randn-5"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randn-6"><span><strong class="def-name">randn</strong> <code class="def-code-arguments">("state", <var class="var">v</var>)</code><a class="copiable-link" href="#index-randn-6"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randn-7"><span><strong class="def-name">randn</strong> <code class="def-code-arguments">("state", "reset")</code><a class="copiable-link" href="#index-randn-7"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randn-8"><span><code class="def-type"><var class="var">v</var> =</code> <strong class="def-name">randn</strong> <code class="def-code-arguments">("seed")</code><a class="copiable-link" href="#index-randn-8"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randn-9"><span><strong class="def-name">randn</strong> <code class="def-code-arguments">("seed", <var class="var">v</var>)</code><a class="copiable-link" href="#index-randn-9"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randn-10"><span><strong class="def-name">randn</strong> <code class="def-code-arguments">("seed", "reset")</code><a class="copiable-link" href="#index-randn-10"> ¶</a></span></dt>
<dd><p>Return a matrix with normally distributed random elements having zero mean
and variance one.
</p>
<p>The arguments are handled the same as the arguments for <code class="code">rand</code>.
</p>
<p>By default, <code class="code">randn</code> uses the Marsaglia and Tsang
“Ziggurat technique” to transform from a uniform to a normal distribution.
</p>
<p>The class of the value returned can be controlled by a trailing
<code class="code">"double"</code> or <code class="code">"single"</code> argument. These are the only valid
classes.
</p>
<p>Reference: G. Marsaglia and W.W. Tsang,
<cite class="cite">Ziggurat Method for Generating Random Variables</cite>,
J. Statistical Software, vol 5, 2000,
<a class="url" href="https://www.jstatsoft.org/v05/i08/">https://www.jstatsoft.org/v05/i08/</a>
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFrand">rand</a>, <a class="ref" href="#XREFrande">rande</a>, <a class="ref" href="#XREFrandg">randg</a>, <a class="ref" href="#XREFrandp">randp</a>.
</p></dd></dl>
<a class="anchor" id="XREFrande"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-rande"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">rande</strong> <code class="def-code-arguments">(<var class="var">n</var>)</code><a class="copiable-link" href="#index-rande"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rande-1"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">rande</strong> <code class="def-code-arguments">(<var class="var">m</var>, <var class="var">n</var>, …)</code><a class="copiable-link" href="#index-rande-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rande-2"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">rande</strong> <code class="def-code-arguments">([<var class="var">m</var> <var class="var">n</var> …])</code><a class="copiable-link" href="#index-rande-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rande-3"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">rande</strong> <code class="def-code-arguments">(…, "single")</code><a class="copiable-link" href="#index-rande-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rande-4"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">rande</strong> <code class="def-code-arguments">(…, "double")</code><a class="copiable-link" href="#index-rande-4"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rande-5"><span><code class="def-type"><var class="var">v</var> =</code> <strong class="def-name">rande</strong> <code class="def-code-arguments">("state")</code><a class="copiable-link" href="#index-rande-5"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rande-6"><span><strong class="def-name">rande</strong> <code class="def-code-arguments">("state", <var class="var">v</var>)</code><a class="copiable-link" href="#index-rande-6"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rande-7"><span><strong class="def-name">rande</strong> <code class="def-code-arguments">("state", "reset")</code><a class="copiable-link" href="#index-rande-7"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rande-8"><span><code class="def-type"><var class="var">v</var> =</code> <strong class="def-name">rande</strong> <code class="def-code-arguments">("seed")</code><a class="copiable-link" href="#index-rande-8"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rande-9"><span><strong class="def-name">rande</strong> <code class="def-code-arguments">("seed", <var class="var">v</var>)</code><a class="copiable-link" href="#index-rande-9"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rande-10"><span><strong class="def-name">rande</strong> <code class="def-code-arguments">("seed", "reset")</code><a class="copiable-link" href="#index-rande-10"> ¶</a></span></dt>
<dd><p>Return a matrix with exponentially distributed random elements.
</p>
<p>The arguments are handled the same as the arguments for <code class="code">rand</code>.
</p>
<p>By default, <code class="code">rande</code> uses the Marsaglia and Tsang
“Ziggurat technique” to transform from a uniform to an exponential
distribution.
</p>
<p>The class of the value returned can be controlled by a trailing
<code class="code">"double"</code> or <code class="code">"single"</code> argument. These are the only valid
classes.
</p>
<p>Reference: G. Marsaglia and W.W. Tsang,
<cite class="cite">Ziggurat Method for Generating Random Variables</cite>,
J. Statistical Software, vol 5, 2000,
<a class="url" href="https://www.jstatsoft.org/v05/i08/">https://www.jstatsoft.org/v05/i08/</a>
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFrand">rand</a>, <a class="ref" href="#XREFrandn">randn</a>, <a class="ref" href="#XREFrandg">randg</a>, <a class="ref" href="#XREFrandp">randp</a>.
</p></dd></dl>
<a class="anchor" id="XREFrandp"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-randp"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">randp</strong> <code class="def-code-arguments">(<var class="var">l</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-randp"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randp-1"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">randp</strong> <code class="def-code-arguments">(<var class="var">l</var>, <var class="var">m</var>, <var class="var">n</var>, …)</code><a class="copiable-link" href="#index-randp-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randp-2"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">randp</strong> <code class="def-code-arguments">(<var class="var">l</var>, [<var class="var">m</var> <var class="var">n</var> …])</code><a class="copiable-link" href="#index-randp-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randp-3"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">randp</strong> <code class="def-code-arguments">(…, "single")</code><a class="copiable-link" href="#index-randp-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randp-4"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">randp</strong> <code class="def-code-arguments">(…, "double")</code><a class="copiable-link" href="#index-randp-4"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randp-5"><span><code class="def-type"><var class="var">v</var> =</code> <strong class="def-name">randp</strong> <code class="def-code-arguments">("state")</code><a class="copiable-link" href="#index-randp-5"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randp-6"><span><strong class="def-name">randp</strong> <code class="def-code-arguments">("state", <var class="var">v</var>)</code><a class="copiable-link" href="#index-randp-6"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randp-7"><span><strong class="def-name">randp</strong> <code class="def-code-arguments">("state", "reset")</code><a class="copiable-link" href="#index-randp-7"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randp-8"><span><code class="def-type"><var class="var">v</var> =</code> <strong class="def-name">randp</strong> <code class="def-code-arguments">("seed")</code><a class="copiable-link" href="#index-randp-8"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randp-9"><span><strong class="def-name">randp</strong> <code class="def-code-arguments">("seed", <var class="var">v</var>)</code><a class="copiable-link" href="#index-randp-9"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randp-10"><span><strong class="def-name">randp</strong> <code class="def-code-arguments">("seed", "reset")</code><a class="copiable-link" href="#index-randp-10"> ¶</a></span></dt>
<dd><p>Return a matrix with Poisson distributed random elements with mean value
parameter given by the first argument, <var class="var">l</var>.
</p>
<p>The arguments are handled the same as the arguments for <code class="code">rand</code>, except
for the argument <var class="var">l</var>.
</p>
<p>Five different algorithms are used depending on the range of <var class="var">l</var> and
whether or not <var class="var">l</var> is a scalar or a matrix.
</p>
<dl class="table">
<dt>For scalar <var class="var">l</var> ≤ 12, use direct method.</dt>
<dd><p>W.H. Press, et al., <cite class="cite">Numerical Recipes in C</cite>,
Cambridge University Press, 1992.
</p>
</dd>
<dt>For scalar <var class="var">l</var> > 12, use rejection method.[1]</dt>
<dd><p>W.H. Press, et al., <cite class="cite">Numerical Recipes in C</cite>,
Cambridge University Press, 1992.
</p>
</dd>
<dt>For matrix <var class="var">l</var> ≤ 10, use inversion method.[2]</dt>
<dd><p>E. Stadlober, et al., WinRand source code, available via FTP.
</p>
</dd>
<dt>For matrix <var class="var">l</var> > 10, use patchwork rejection method.</dt>
<dd><p>E. Stadlober, et al., WinRand source code, available via FTP, or
H. Zechner, <cite class="cite">Efficient sampling from continuous and discrete
unimodal distributions</cite>, Doctoral Dissertation, 156pp., Technical
University Graz, Austria, 1994.
</p>
</dd>
<dt>For <var class="var">l</var> > 1e8, use normal approximation.</dt>
<dd><p>L. Montanet, et al., <cite class="cite">Review of Particle Properties</cite>,
Physical Review D 50 p1284, 1994.
</p></dd>
</dl>
<p>The class of the value returned can be controlled by a trailing
<code class="code">"double"</code> or <code class="code">"single"</code> argument. These are the only valid
classes.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFrand">rand</a>, <a class="ref" href="#XREFrandn">randn</a>, <a class="ref" href="#XREFrande">rande</a>, <a class="ref" href="#XREFrandg">randg</a>.
</p></dd></dl>
<a class="anchor" id="XREFrandg"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-randg"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">randg</strong> <code class="def-code-arguments">(<var class="var">a</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-randg"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randg-1"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">randg</strong> <code class="def-code-arguments">(<var class="var">a</var>, <var class="var">m</var>, <var class="var">n</var>, …)</code><a class="copiable-link" href="#index-randg-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randg-2"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">randg</strong> <code class="def-code-arguments">(<var class="var">a</var>, [<var class="var">m</var> <var class="var">n</var> …])</code><a class="copiable-link" href="#index-randg-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randg-3"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">randg</strong> <code class="def-code-arguments">(…, "single")</code><a class="copiable-link" href="#index-randg-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randg-4"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">randg</strong> <code class="def-code-arguments">(…, "double")</code><a class="copiable-link" href="#index-randg-4"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randg-5"><span><code class="def-type"><var class="var">v</var> =</code> <strong class="def-name">randg</strong> <code class="def-code-arguments">("state")</code><a class="copiable-link" href="#index-randg-5"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randg-6"><span><strong class="def-name">randg</strong> <code class="def-code-arguments">("state", <var class="var">v</var>)</code><a class="copiable-link" href="#index-randg-6"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randg-7"><span><strong class="def-name">randg</strong> <code class="def-code-arguments">("state", "reset")</code><a class="copiable-link" href="#index-randg-7"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randg-8"><span><code class="def-type"><var class="var">v</var> =</code> <strong class="def-name">randg</strong> <code class="def-code-arguments">("seed")</code><a class="copiable-link" href="#index-randg-8"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randg-9"><span><strong class="def-name">randg</strong> <code class="def-code-arguments">("seed", <var class="var">v</var>)</code><a class="copiable-link" href="#index-randg-9"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randg-10"><span><strong class="def-name">randg</strong> <code class="def-code-arguments">("seed", "reset")</code><a class="copiable-link" href="#index-randg-10"> ¶</a></span></dt>
<dd>
<p>Return a matrix with <code class="code">gamma (<var class="var">a</var>,1)</code> distributed random elements.
</p>
<p>The arguments are handled the same as the arguments for <code class="code">rand</code>, except
for the argument <var class="var">a</var>.
</p>
<p>This can be used to generate many distributions:
</p>
<dl class="table">
<dt><code class="code">gamma (a, b)</code> for <code class="code">a > -1</code>, <code class="code">b > 0</code></dt>
<dd>
<div class="example">
<pre class="example-preformatted">r = b * randg (a)
</pre></div>
</dd>
<dt><code class="code">beta (a, b)</code> for <code class="code">a > -1</code>, <code class="code">b > -1</code></dt>
<dd>
<div class="example">
<div class="group"><pre class="example-preformatted">r1 = randg (a, 1)
r = r1 / (r1 + randg (b, 1))
</pre></div></div>
</dd>
<dt><code class="code">Erlang (a, n)</code></dt>
<dd>
<div class="example">
<pre class="example-preformatted">r = a * randg (n)
</pre></div>
</dd>
<dt><code class="code">chisq (df)</code> for <code class="code">df > 0</code></dt>
<dd>
<div class="example">
<pre class="example-preformatted">r = 2 * randg (df / 2)
</pre></div>
</dd>
<dt><code class="code">t (df)</code> for <code class="code">0 < df < inf</code> (use randn if df is infinite)</dt>
<dd>
<div class="example">
<pre class="example-preformatted">r = randn () / sqrt (2 * randg (df / 2) / df)
</pre></div>
</dd>
<dt><code class="code">F (n1, n2)</code> for <code class="code">0 < n1</code>, <code class="code">0 < n2</code></dt>
<dd>
<div class="example">
<div class="group"><pre class="example-preformatted">## r1 equals 1 if n1 is infinite
r1 = 2 * randg (n1 / 2) / n1
## r2 equals 1 if n2 is infinite
r2 = 2 * randg (n2 / 2) / n2
r = r1 / r2
</pre></div></div>
</dd>
<dt>negative <code class="code">binomial (n, p)</code> for <code class="code">n > 0</code>, <code class="code">0 < p <= 1</code></dt>
<dd>
<div class="example">
<pre class="example-preformatted">r = randp ((1 - p) / p * randg (n))
</pre></div>
</dd>
<dt>non-central <code class="code">chisq (df, L)</code>, for <code class="code">df >= 0</code> and <code class="code">L > 0</code></dt>
<dd><p>(use chisq if <code class="code">L = 0</code>)
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">r = randp (L / 2)
r(r > 0) = 2 * randg (r(r > 0))
r(df > 0) += 2 * randg (df(df > 0)/2)
</pre></div></div>
</dd>
<dt><code class="code">Dirichlet (a1, … ak)</code></dt>
<dd>
<div class="example">
<div class="group"><pre class="example-preformatted">r = (randg (a1), ..., randg (ak))
r = r / sum (r)
</pre></div></div>
</dd>
</dl>
<p>The class of the value returned can be controlled by a trailing
<code class="code">"double"</code> or <code class="code">"single"</code> argument. These are the only valid
classes.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFrand">rand</a>, <a class="ref" href="#XREFrandn">randn</a>, <a class="ref" href="#XREFrande">rande</a>, <a class="ref" href="#XREFrandp">randp</a>.
</p></dd></dl>
<a class="anchor" id="XREFrng"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-rng"><span><strong class="def-name">rng</strong> <code class="def-code-arguments">(<var class="var">seed</var>)</code><a class="copiable-link" href="#index-rng"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rng-1"><span><strong class="def-name">rng</strong> <code class="def-code-arguments">(<var class="var">seed</var>, "<var class="var">generator</var>")</code><a class="copiable-link" href="#index-rng-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rng-2"><span><strong class="def-name">rng</strong> <code class="def-code-arguments">("shuffle")</code><a class="copiable-link" href="#index-rng-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rng-3"><span><strong class="def-name">rng</strong> <code class="def-code-arguments">("shuffle", "<var class="var">generator</var>")</code><a class="copiable-link" href="#index-rng-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rng-4"><span><strong class="def-name">rng</strong> <code class="def-code-arguments">("default")</code><a class="copiable-link" href="#index-rng-4"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rng-5"><span><code class="def-type"><var class="var">s</var> =</code> <strong class="def-name">rng</strong> <code class="def-code-arguments">()</code><a class="copiable-link" href="#index-rng-5"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rng-6"><span><strong class="def-name">rng</strong> <code class="def-code-arguments">(<var class="var">s</var>)</code><a class="copiable-link" href="#index-rng-6"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rng-7"><span><code class="def-type"><var class="var">s</var> =</code> <strong class="def-name">rng</strong> <code class="def-code-arguments">(…)</code><a class="copiable-link" href="#index-rng-7"> ¶</a></span></dt>
<dd><p>Set or query the seed of the random number generator used by <code class="code">rand</code> and
<code class="code">randn</code>.
</p>
<p>The input <var class="var">seed</var> is a scalar numeric value used to initialize the state
vector of the random number generator.
</p>
<p>The optional string <var class="var">generator</var> specifies the type of random number
generator to be used. Its value can be <code class="code">"twister"</code>,
<code class="code">"v5uniform"</code>, or <code class="code">"v5normal"</code>. The <code class="code">"twister"</code> keyword
is described below. <code class="code">"v5uniform"</code> and <code class="code">"v5normal"</code> refer to
older versions of Octave that used to use a different random number
generator.
</p>
<p>The state or seed of the random number generator can be reset to a new
random value using the <code class="code">"shuffle"</code> keyword.
</p>
<p>The random number generator can be reset to default values using the
<code class="code">"default"</code> keyword. The default values are to use the Mersenne
Twister generator with a seed of 0.
</p>
<p>The optional return value <var class="var">s</var> contains the state of the random number
generator at the time the function is called (i.e., before it might be
modified according to the input arguments). It is encoded as a structure
variable with three fields: <code class="code">"Type"</code>, <code class="code">"Seed"</code>, and
<code class="code">"State"</code>. The random number generator can be restored to the state
<var class="var">s</var> using <code class="code">rng (<var class="var">s</var>)</code>. This is useful when the identical
sequence of pseudo-random numbers is required for an algorithm.
</p>
<p>By default, and with the <code class="code">"twister"</code> option, pseudo-random sequences
are computed using the Mersenne Twister with a period of <em class="math">2^{19937}-1</em>
(See M. Matsumoto and T. Nishimura,
<cite class="cite">Mersenne Twister: A 623-dimensionally equidistributed uniform
pseudorandom number generator</cite>,
ACM Trans. on Modeling and Computer Simulation Vol. 8, No. 1,
pp. 3–30, January 1998,
<a class="url" href="http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html">http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html</a>).
Do <strong class="strong">not</strong> use for cryptography without securely hashing several
returned values together, otherwise the generator state can be learned after
reading 624 consecutive values.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFrand">rand</a>, <a class="ref" href="#XREFrandn">randn</a>.
</p></dd></dl>
<p>The generators operate in the new or old style together, it is not
possible to mix the two. Initializing any generator with
<code class="code">"state"</code> or <code class="code">"seed"</code> causes the others to switch to the
same style for future calls.
</p>
<p>The state of each generator is independent and calls to different
generators can be interleaved without affecting the final result. For
example,
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">rand ("state", [11, 22, 33]);
randn ("state", [44, 55, 66]);
u = rand (100, 1);
n = randn (100, 1);
</pre></div></div>
<p>and
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">rand ("state", [11, 22, 33]);
randn ("state", [44, 55, 66]);
u = zeros (100, 1);
n = zeros (100, 1);
for i = 1:100
u(i) = rand ();
n(i) = randn ();
end
</pre></div></div>
<p>produce equivalent results. When the generators are initialized in
the old style with <code class="code">"seed"</code> only <code class="code">rand</code> and <code class="code">randn</code> are
independent, because the old <code class="code">rande</code>, <code class="code">randg</code> and
<code class="code">randp</code> generators make calls to <code class="code">rand</code> and <code class="code">randn</code>.
</p>
<p>The generators are initialized with random states at start-up, so
that the sequences of random numbers are not the same each time you run
Octave.<a class="footnote" id="DOCF7" href="#FOOT7"><sup>7</sup></a> If you really do
need to reproduce a sequence of numbers exactly, you can set the state
or seed to a specific value.
</p>
<p>If invoked without arguments, <code class="code">rand</code> and <code class="code">randn</code> return a
single element of a random sequence.
</p>
<p>The original <code class="code">rand</code> and <code class="code">randn</code> functions use Fortran code from
<small class="sc">RANLIB</small>, a library of Fortran routines for random number generation,
compiled by Barry W. Brown and James Lovato of the Department of
Biomathematics at The University of Texas, M.D. Anderson Cancer Center,
Houston, TX 77030.
</p>
<a class="anchor" id="XREFrandperm"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-randperm"><span><code class="def-type"><var class="var">v</var> =</code> <strong class="def-name">randperm</strong> <code class="def-code-arguments">(<var class="var">n</var>)</code><a class="copiable-link" href="#index-randperm"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-randperm-1"><span><code class="def-type"><var class="var">v</var> =</code> <strong class="def-name">randperm</strong> <code class="def-code-arguments">(<var class="var">n</var>, <var class="var">m</var>)</code><a class="copiable-link" href="#index-randperm-1"> ¶</a></span></dt>
<dd><p>Return a row vector containing a random permutation of <code class="code">1:<var class="var">n</var></code>.
</p>
<p>If <var class="var">m</var> is supplied, return <var class="var">m</var> unique entries, sampled without
replacement from <code class="code">1:<var class="var">n</var></code>.
</p>
<p>The complexity is O(<var class="var">n</var>) in memory and O(<var class="var">m</var>) in time, unless
<var class="var">m</var> < <var class="var">n</var>/5, in which case O(<var class="var">m</var>) memory is used as well. The
randomization is performed using rand(). All permutations are equally
likely.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="Basic-Statistical-Functions.html#XREFperms">perms</a>.
</p></dd></dl>
</div>
<div class="footnotes-segment">
<hr>
<h4 class="footnotes-heading">Footnotes</h4>
<h5 class="footnote-body-heading"><a id="FOOT7" href="#DOCF7">(7)</a></h5>
<p>The old versions of <code class="code">rand</code> and <code class="code">randn</code>
obtain their initial seeds from the system clock.</p>
</div>
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