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<a name="Utility-Functions"></a>
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<p>
Next: <a href="Special-Functions.html#Special-Functions" accesskey="n" rel="next">Special Functions</a>, Previous: <a href="Sums-and-Products.html#Sums-and-Products" accesskey="p" rel="prev">Sums and Products</a>, Up: <a href="Arithmetic.html#Arithmetic" accesskey="u" rel="up">Arithmetic</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Utility-Functions-1"></a>
<h3 class="section">17.5 Utility Functions</h3>
<a name="XREFceil"></a><dl>
<dt><a name="index-ceil"></a><em></em> <strong>ceil</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Return the smallest integer not less than <var>x</var>.
</p>
<p>This is equivalent to rounding towards positive infinity.
</p>
<p>If <var>x</var> is complex, return
<code>ceil (real (<var>x</var>)) + ceil (imag (<var>x</var>)) * I</code>.
</p>
<div class="example">
<pre class="example">ceil ([-2.7, 2.7])
⇒ -2 3
</pre></div>
<p><strong>See also:</strong> <a href="#XREFfloor">floor</a>, <a href="#XREFround">round</a>, <a href="#XREFfix">fix</a>.
</p></dd></dl>
<a name="XREFfix"></a><dl>
<dt><a name="index-fix"></a><em></em> <strong>fix</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Truncate fractional portion of <var>x</var> and return the integer portion.
</p>
<p>This is equivalent to rounding towards zero. If <var>x</var> is complex, return
<code>fix (real (<var>x</var>)) + fix (imag (<var>x</var>)) * I</code>.
</p>
<div class="example">
<pre class="example">fix ([-2.7, 2.7])
⇒ -2 2
</pre></div>
<p><strong>See also:</strong> <a href="#XREFceil">ceil</a>, <a href="#XREFfloor">floor</a>, <a href="#XREFround">round</a>.
</p></dd></dl>
<a name="XREFfloor"></a><dl>
<dt><a name="index-floor"></a><em></em> <strong>floor</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Return the largest integer not greater than <var>x</var>.
</p>
<p>This is equivalent to rounding towards negative infinity. If <var>x</var> is
complex, return <code>floor (real (<var>x</var>)) + floor (imag (<var>x</var>)) * I</code>.
</p>
<div class="example">
<pre class="example">floor ([-2.7, 2.7])
⇒ -3 2
</pre></div>
<p><strong>See also:</strong> <a href="#XREFceil">ceil</a>, <a href="#XREFround">round</a>, <a href="#XREFfix">fix</a>.
</p></dd></dl>
<a name="XREFround"></a><dl>
<dt><a name="index-round"></a><em></em> <strong>round</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Return the integer nearest to <var>x</var>.
</p>
<p>If <var>x</var> is complex, return
<code>round (real (<var>x</var>)) + round (imag (<var>x</var>)) * I</code>. If there
are two nearest integers, return the one further away from zero.
</p>
<div class="example">
<pre class="example">round ([-2.7, 2.7])
⇒ -3 3
</pre></div>
<p><strong>See also:</strong> <a href="#XREFceil">ceil</a>, <a href="#XREFfloor">floor</a>, <a href="#XREFfix">fix</a>, <a href="#XREFroundb">roundb</a>.
</p></dd></dl>
<a name="XREFroundb"></a><dl>
<dt><a name="index-roundb"></a><em></em> <strong>roundb</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Return the integer nearest to <var>x</var>. If there are two nearest
integers, return the even one (banker’s rounding).
</p>
<p>If <var>x</var> is complex,
return <code>roundb (real (<var>x</var>)) + roundb (imag (<var>x</var>)) * I</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFround">round</a>.
</p></dd></dl>
<a name="XREFmax"></a><dl>
<dt><a name="index-max"></a><em></em> <strong>max</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-max-1"></a><em></em> <strong>max</strong> <em>(<var>x</var>, [], <var>dim</var>)</em></dt>
<dt><a name="index-max-2"></a><em>[<var>w</var>, <var>iw</var>] =</em> <strong>max</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-max-3"></a><em></em> <strong>max</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dd><p>Find maximum values in the array <var>x</var>.
</p>
<p>For a vector argument, return the maximum value. For a matrix argument,
return a row vector with the maximum value of each column. For a
multi-dimensional array, <code>max</code> operates along the first non-singleton
dimension.
</p>
<p>If the optional third argument <var>dim</var> is present then operate along
this dimension. In this case the second argument is ignored and should be
set to the empty matrix.
</p>
<p>For two matrices (or a matrix and a scalar), return the pairwise maximum.
</p>
<p>Thus,
</p>
<div class="example">
<pre class="example">max (max (<var>x</var>))
</pre></div>
<p>returns the largest element of the 2-D matrix <var>x</var>, and
</p>
<div class="example">
<pre class="example">max (2:5, pi)
⇒ 3.1416 3.1416 4.0000 5.0000
</pre></div>
<p>compares each element of the range <code>2:5</code> with <code>pi</code>, and returns a
row vector of the maximum values.
</p>
<p>For complex arguments, the magnitude of the elements are used for
comparison. If the magnitudes are identical, then the results are ordered
by phase angle in the range (-pi, pi]. Hence,
</p>
<div class="example">
<pre class="example">max ([-1 i 1 -i])
⇒ -1
</pre></div>
<p>because all entries have magnitude 1, but -1 has the largest phase angle
with value pi.
</p>
<p>If called with one input and two output arguments, <code>max</code> also returns
the first index of the maximum value(s). Thus,
</p>
<div class="example">
<pre class="example">[x, ix] = max ([1, 3, 5, 2, 5])
⇒ x = 5
ix = 3
</pre></div>
<p><strong>See also:</strong> <a href="#XREFmin">min</a>, <a href="#XREFcummax">cummax</a>, <a href="#XREFcummin">cummin</a>.
</p></dd></dl>
<a name="XREFmin"></a><dl>
<dt><a name="index-min"></a><em></em> <strong>min</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-min-1"></a><em></em> <strong>min</strong> <em>(<var>x</var>, [], <var>dim</var>)</em></dt>
<dt><a name="index-min-2"></a><em>[<var>w</var>, <var>iw</var>] =</em> <strong>min</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-min-3"></a><em></em> <strong>min</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dd><p>Find minimum values in the array <var>x</var>.
</p>
<p>For a vector argument, return the minimum value. For a matrix argument,
return a row vector with the minimum value of each column. For a
multi-dimensional array, <code>min</code> operates along the first non-singleton
dimension.
</p>
<p>If the optional third argument <var>dim</var> is present then operate along
this dimension. In this case the second argument is ignored and should be
set to the empty matrix.
</p>
<p>For two matrices (or a matrix and a scalar), return the pairwise minimum.
</p>
<p>Thus,
</p>
<div class="example">
<pre class="example">min (min (<var>x</var>))
</pre></div>
<p>returns the smallest element of the 2-D matrix <var>x</var>, and
</p>
<div class="example">
<pre class="example">min (2:5, pi)
⇒ 2.0000 3.0000 3.1416 3.1416
</pre></div>
<p>compares each element of the range <code>2:5</code> with <code>pi</code>, and returns a
row vector of the minimum values.
</p>
<p>For complex arguments, the magnitude of the elements are used for
comparison. If the magnitudes are identical, then the results are ordered
by phase angle in the range (-pi, pi]. Hence,
</p>
<div class="example">
<pre class="example">min ([-1 i 1 -i])
⇒ -i
</pre></div>
<p>because all entries have magnitude 1, but -i has the smallest phase angle
with value -pi/2.
</p>
<p>If called with one input and two output arguments, <code>min</code> also returns
the first index of the minimum value(s). Thus,
</p>
<div class="example">
<pre class="example">[x, ix] = min ([1, 3, 0, 2, 0])
⇒ x = 0
ix = 3
</pre></div>
<p><strong>See also:</strong> <a href="#XREFmax">max</a>, <a href="#XREFcummin">cummin</a>, <a href="#XREFcummax">cummax</a>.
</p></dd></dl>
<a name="XREFcummax"></a><dl>
<dt><a name="index-cummax"></a><em></em> <strong>cummax</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-cummax-1"></a><em></em> <strong>cummax</strong> <em>(<var>x</var>, <var>dim</var>)</em></dt>
<dt><a name="index-cummax-2"></a><em>[<var>w</var>, <var>iw</var>] =</em> <strong>cummax</strong> <em>(…)</em></dt>
<dd><p>Return the cumulative maximum values along dimension <var>dim</var>.
</p>
<p>If <var>dim</var> is unspecified it defaults to column-wise operation. For
example:
</p>
<div class="example">
<pre class="example">cummax ([1 3 2 6 4 5])
⇒ 1 3 3 6 6 6
</pre></div>
<p>If called with two output arguments the index of the maximum value is also
returned.
</p>
<div class="example">
<pre class="example">[w, iw] = cummax ([1 3 2 6 4 5])
⇒
w = 1 3 3 6 6 6
iw = 1 2 2 4 4 4
</pre></div>
<p><strong>See also:</strong> <a href="#XREFcummin">cummin</a>, <a href="#XREFmax">max</a>, <a href="#XREFmin">min</a>.
</p></dd></dl>
<a name="XREFcummin"></a><dl>
<dt><a name="index-cummin"></a><em></em> <strong>cummin</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-cummin-1"></a><em></em> <strong>cummin</strong> <em>(<var>x</var>, <var>dim</var>)</em></dt>
<dt><a name="index-cummin-2"></a><em>[<var>w</var>, <var>iw</var>] =</em> <strong>cummin</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Return the cumulative minimum values along dimension <var>dim</var>.
</p>
<p>If <var>dim</var> is unspecified it defaults to column-wise operation. For
example:
</p>
<div class="example">
<pre class="example">cummin ([5 4 6 2 3 1])
⇒ 5 4 4 2 2 1
</pre></div>
<p>If called with two output arguments the index of the minimum value is also
returned.
</p>
<div class="example">
<pre class="example">[w, iw] = cummin ([5 4 6 2 3 1])
⇒
w = 5 4 4 2 2 1
iw = 1 2 2 4 4 6
</pre></div>
<p><strong>See also:</strong> <a href="#XREFcummax">cummax</a>, <a href="#XREFmin">min</a>, <a href="#XREFmax">max</a>.
</p></dd></dl>
<a name="XREFhypot"></a><dl>
<dt><a name="index-hypot"></a><em></em> <strong>hypot</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dt><a name="index-hypot-1"></a><em></em> <strong>hypot</strong> <em>(<var>x</var>, <var>y</var>, <var>z</var>, …)</em></dt>
<dd><p>Compute the element-by-element square root of the sum of the squares of
<var>x</var> and <var>y</var>.
</p>
<p>This is equivalent to
<code>sqrt (<var>x</var>.^2 + <var>y</var>.^2)</code>, but is calculated in a manner that
avoids overflows for large values of <var>x</var> or <var>y</var>.
</p>
<p><code>hypot</code> can also be called with more than 2 arguments; in this case,
the arguments are accumulated from left to right:
</p>
<div class="example">
<pre class="example">hypot (hypot (<var>x</var>, <var>y</var>), <var>z</var>)
hypot (hypot (hypot (<var>x</var>, <var>y</var>), <var>z</var>), <var>w</var>), etc.
</pre></div>
</dd></dl>
<a name="XREFgradient"></a><dl>
<dt><a name="index-gradient"></a><em><var>dx</var> =</em> <strong>gradient</strong> <em>(<var>m</var>)</em></dt>
<dt><a name="index-gradient-1"></a><em>[<var>dx</var>, <var>dy</var>, <var>dz</var>, …] =</em> <strong>gradient</strong> <em>(<var>m</var>)</em></dt>
<dt><a name="index-gradient-2"></a><em>[…] =</em> <strong>gradient</strong> <em>(<var>m</var>, <var>s</var>)</em></dt>
<dt><a name="index-gradient-3"></a><em>[…] =</em> <strong>gradient</strong> <em>(<var>m</var>, <var>x</var>, <var>y</var>, <var>z</var>, …)</em></dt>
<dt><a name="index-gradient-4"></a><em>[…] =</em> <strong>gradient</strong> <em>(<var>f</var>, <var>x0</var>)</em></dt>
<dt><a name="index-gradient-5"></a><em>[…] =</em> <strong>gradient</strong> <em>(<var>f</var>, <var>x0</var>, <var>s</var>)</em></dt>
<dt><a name="index-gradient-6"></a><em>[…] =</em> <strong>gradient</strong> <em>(<var>f</var>, <var>x0</var>, <var>x</var>, <var>y</var>, …)</em></dt>
<dd>
<p>Calculate the gradient of sampled data or a function.
</p>
<p>If <var>m</var> is a vector, calculate the one-dimensional gradient of <var>m</var>.
If <var>m</var> is a matrix the gradient is calculated for each dimension.
</p>
<p><code>[<var>dx</var>, <var>dy</var>] = gradient (<var>m</var>)</code> calculates the
one-dimensional gradient for <var>x</var> and <var>y</var> direction if <var>m</var> is a
matrix. Additional return arguments can be use for multi-dimensional
matrices.
</p>
<p>A constant spacing between two points can be provided by the <var>s</var>
parameter. If <var>s</var> is a scalar, it is assumed to be the spacing for all
dimensions. Otherwise, separate values of the spacing can be supplied by
the <var>x</var>, … arguments. Scalar values specify an equidistant
spacing. Vector values for the <var>x</var>, … arguments specify the
coordinate for that dimension. The length must match their respective
dimension of <var>m</var>.
</p>
<p>At boundary points a linear extrapolation is applied. Interior points
are calculated with the first approximation of the numerical gradient
</p>
<div class="example">
<pre class="example">y'(i) = 1/(x(i+1)-x(i-1)) * (y(i-1)-y(i+1)).
</pre></div>
<p>If the first argument <var>f</var> is a function handle, the gradient of the
function at the points in <var>x0</var> is approximated using central difference.
For example, <code>gradient (@cos, 0)</code> approximates the gradient of the
cosine function in the point <em>x0 = 0</em>. As with sampled data, the
spacing values between the points from which the gradient is estimated can
be set via the <var>s</var> or <var>dx</var>, <var>dy</var>, … arguments. By default
a spacing of 1 is used.
</p>
<p><strong>See also:</strong> <a href="Finding-Elements-and-Checking-Conditions.html#XREFdiff">diff</a>, <a href="#XREFdel2">del2</a>.
</p></dd></dl>
<a name="XREFdot"></a><dl>
<dt><a name="index-dot"></a><em></em> <strong>dot</strong> <em>(<var>x</var>, <var>y</var>, <var>dim</var>)</em></dt>
<dd><p>Compute the dot product of two vectors.
</p>
<p>If <var>x</var> and <var>y</var> are matrices, calculate the dot products along the
first non-singleton dimension.
</p>
<p>If the optional argument <var>dim</var> is given, calculate the dot products
along this dimension.
</p>
<p>This is equivalent to
<code>sum (conj (<var>X</var>) .* <var>Y</var>, <var>dim</var>)</code>,
but avoids forming a temporary array and is faster. When <var>X</var> and
<var>Y</var> are column vectors, the result is equivalent to
<code><var>X</var>' * <var>Y</var></code>.
</p>
<p><strong>See also:</strong> <a href="#XREFcross">cross</a>, <a href="#XREFdivergence">divergence</a>.
</p></dd></dl>
<a name="XREFcross"></a><dl>
<dt><a name="index-cross"></a><em></em> <strong>cross</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dt><a name="index-cross-1"></a><em></em> <strong>cross</strong> <em>(<var>x</var>, <var>y</var>, <var>dim</var>)</em></dt>
<dd><p>Compute the vector cross product of two 3-dimensional vectors <var>x</var> and
<var>y</var>.
</p>
<p>If <var>x</var> and <var>y</var> are matrices, the cross product is applied along the
first dimension with three elements.
</p>
<p>The optional argument <var>dim</var> forces the cross product to be calculated
along the specified dimension.
</p>
<p>Example Code:
</p>
<div class="example">
<pre class="example">cross ([1, 1, 0], [0, 1, 1])
⇒
1 -1 1
</pre></div>
<p><strong>See also:</strong> <a href="#XREFdot">dot</a>, <a href="#XREFcurl">curl</a>, <a href="#XREFdivergence">divergence</a>.
</p></dd></dl>
<a name="XREFdivergence"></a><dl>
<dt><a name="index-divergence"></a><em><var>div</var> =</em> <strong>divergence</strong> <em>(<var>x</var>, <var>y</var>, <var>z</var>, <var>fx</var>, <var>fy</var>, <var>fz</var>)</em></dt>
<dt><a name="index-divergence-1"></a><em><var>div</var> =</em> <strong>divergence</strong> <em>(<var>fx</var>, <var>fy</var>, <var>fz</var>)</em></dt>
<dt><a name="index-divergence-2"></a><em><var>div</var> =</em> <strong>divergence</strong> <em>(<var>x</var>, <var>y</var>, <var>fx</var>, <var>fy</var>)</em></dt>
<dt><a name="index-divergence-3"></a><em><var>div</var> =</em> <strong>divergence</strong> <em>(<var>fx</var>, <var>fy</var>)</em></dt>
<dd><p>Calculate divergence of a vector field given by the arrays <var>fx</var>,
<var>fy</var>, and <var>fz</var> or <var>fx</var>, <var>fy</var> respectively.
</p>
<div class="example">
<pre class="example"> d d d
div F(x,y,z) = -- F(x,y,z) + -- F(x,y,z) + -- F(x,y,z)
dx dy dz
</pre></div>
<p>The coordinates of the vector field can be given by the arguments <var>x</var>,
<var>y</var>, <var>z</var> or <var>x</var>, <var>y</var> respectively.
</p>
<p><strong>See also:</strong> <a href="#XREFcurl">curl</a>, <a href="#XREFgradient">gradient</a>, <a href="#XREFdel2">del2</a>, <a href="#XREFdot">dot</a>.
</p></dd></dl>
<a name="XREFcurl"></a><dl>
<dt><a name="index-curl"></a><em>[<var>cx</var>, <var>cy</var>, <var>cz</var>, <var>v</var>] =</em> <strong>curl</strong> <em>(<var>x</var>, <var>y</var>, <var>z</var>, <var>fx</var>, <var>fy</var>, <var>fz</var>)</em></dt>
<dt><a name="index-curl-1"></a><em>[<var>cz</var>, <var>v</var>] =</em> <strong>curl</strong> <em>(<var>x</var>, <var>y</var>, <var>fx</var>, <var>fy</var>)</em></dt>
<dt><a name="index-curl-2"></a><em>[…] =</em> <strong>curl</strong> <em>(<var>fx</var>, <var>fy</var>, <var>fz</var>)</em></dt>
<dt><a name="index-curl-3"></a><em>[…] =</em> <strong>curl</strong> <em>(<var>fx</var>, <var>fy</var>)</em></dt>
<dt><a name="index-curl-4"></a><em><var>v</var> =</em> <strong>curl</strong> <em>(…)</em></dt>
<dd><p>Calculate curl of vector field given by the arrays <var>fx</var>, <var>fy</var>, and
<var>fz</var> or <var>fx</var>, <var>fy</var> respectively.
</p>
<div class="example">
<pre class="example"> / d d d d d d \
curl F(x,y,z) = | -- Fz - -- Fy, -- Fx - -- Fz, -- Fy - -- Fx |
\ dy dz dz dx dx dy /
</pre></div>
<p>The coordinates of the vector field can be given by the arguments <var>x</var>,
<var>y</var>, <var>z</var> or <var>x</var>, <var>y</var> respectively. <var>v</var> calculates the
scalar component of the angular velocity vector in direction of the z-axis
for two-dimensional input. For three-dimensional input the scalar
rotation is calculated at each grid point in direction of the vector field
at that point.
</p>
<p><strong>See also:</strong> <a href="#XREFdivergence">divergence</a>, <a href="#XREFgradient">gradient</a>, <a href="#XREFdel2">del2</a>, <a href="#XREFcross">cross</a>.
</p></dd></dl>
<a name="XREFdel2"></a><dl>
<dt><a name="index-del2"></a><em><var>L</var> =</em> <strong>del2</strong> <em>(<var>M</var>)</em></dt>
<dt><a name="index-del2-1"></a><em><var>L</var> =</em> <strong>del2</strong> <em>(<var>M</var>, <var>h</var>)</em></dt>
<dt><a name="index-del2-2"></a><em><var>L</var> =</em> <strong>del2</strong> <em>(<var>M</var>, <var>dx</var>, <var>dy</var>, …)</em></dt>
<dd>
<p>Calculate the discrete Laplace
operator.
</p>
<p>For a 2-dimensional matrix <var>M</var> this is defined as
</p>
<div class="example">
<pre class="example"> 1 / d^2 d^2 \
L = --- * | --- M(x,y) + --- M(x,y) |
4 \ dx^2 dy^2 /
</pre></div>
<p>For N-dimensional arrays the sum in parentheses is expanded to include
second derivatives over the additional higher dimensions.
</p>
<p>The spacing between evaluation points may be defined by <var>h</var>, which is a
scalar defining the equidistant spacing in all dimensions. Alternatively,
the spacing in each dimension may be defined separately by <var>dx</var>,
<var>dy</var>, etc. A scalar spacing argument defines equidistant spacing,
whereas a vector argument can be used to specify variable spacing. The
length of the spacing vectors must match the respective dimension of
<var>M</var>. The default spacing value is 1.
</p>
<p>Dimensions with fewer than 3 data points are skipped. Boundary points are
calculated from the linear extrapolation of interior points.
</p>
<p>Example: Second derivative of 2*x^3
</p>
<div class="example">
<pre class="example">f = @(x) 2*x.^3;
dd = @(x) 12*x;
x = 1:6;
L = 4*del2 (f(x));
assert (L, dd (x));
</pre></div>
<p><strong>See also:</strong> <a href="#XREFgradient">gradient</a>, <a href="Finding-Elements-and-Checking-Conditions.html#XREFdiff">diff</a>.
</p></dd></dl>
<a name="XREFfactorial"></a><dl>
<dt><a name="index-factorial"></a><em></em> <strong>factorial</strong> <em>(<var>n</var>)</em></dt>
<dd><p>Return the factorial of <var>n</var> where <var>n</var> is a real non-negative
integer.
</p>
<p>If <var>n</var> is a scalar, this is equivalent to <code>prod (1:<var>n</var>)</code>. For
vector or matrix arguments, return the factorial of each element in the
array.
</p>
<p>For non-integers see the generalized factorial function <code>gamma</code>.
Note that the factorial function grows large quite quickly, and even
with double precision values overflow will occur if <var>n</var> > 171. For
such cases consider <code>gammaln</code>.
</p>
<p><strong>See also:</strong> <a href="Sums-and-Products.html#XREFprod">prod</a>, <a href="Special-Functions.html#XREFgamma">gamma</a>, <a href="Special-Functions.html#XREFgammaln">gammaln</a>.
</p></dd></dl>
<a name="XREFfactor"></a><dl>
<dt><a name="index-factor"></a><em><var>pf</var> =</em> <strong>factor</strong> <em>(<var>q</var>)</em></dt>
<dt><a name="index-factor-1"></a><em>[<var>pf</var>, <var>n</var>] =</em> <strong>factor</strong> <em>(<var>q</var>)</em></dt>
<dd><p>Return the prime factorization of <var>q</var>.
</p>
<p>The prime factorization is defined as <code>prod (<var>pf</var>) == <var>q</var></code>
where every element of <var>pf</var> is a prime number. If <code><var>q</var> == 1</code>,
return 1. The output <var>pf</var> is of the same numeric class as the input.
</p>
<p>With two output arguments, return the unique prime factors <var>pf</var> and
their multiplicities. That is,
<code>prod (<var>pf</var> .^ <var>n</var>) == <var>q</var></code>.
</p>
<p>Implementation Note: The input <var>q</var> must be less than <code>flintmax</code>
(9.0072e+15) in order to factor correctly.
</p>
<p><strong>See also:</strong> <a href="#XREFgcd">gcd</a>, <a href="#XREFlcm">lcm</a>, <a href="Predicates-for-Numeric-Objects.html#XREFisprime">isprime</a>, <a href="#XREFprimes">primes</a>.
</p></dd></dl>
<a name="XREFgcd"></a><dl>
<dt><a name="index-gcd"></a><em><var>g</var> =</em> <strong>gcd</strong> <em>(<var>a1</var>, <var>a2</var>, …)</em></dt>
<dt><a name="index-gcd-1"></a><em>[<var>g</var>, <var>v1</var>, …] =</em> <strong>gcd</strong> <em>(<var>a1</var>, <var>a2</var>, …)</em></dt>
<dd><p>Compute the greatest common divisor of <var>a1</var>, <var>a2</var>, ….
</p>
<p>If more than one argument is given then all arguments must be the same size
or scalar. In this case the greatest common divisor is calculated for each
element individually. All elements must be ordinary or Gaussian (complex)
integers. Note that for Gaussian integers, the gcd is only unique up to a
phase factor (multiplication by 1, -1, i, or -i), so an arbitrary greatest
common divisor among the four possible is returned.
</p>
<p>Optional return arguments <var>v1</var>, …, contain integer vectors such
that,
</p>
<div class="example">
<pre class="example"><var>g</var> = <var>v1</var> .* <var>a1</var> + <var>v2</var> .* <var>a2</var> + …
</pre></div>
<p>Example code:
</p>
<div class="example">
<pre class="example">gcd ([15, 9], [20, 18])
⇒ 5 9
</pre></div>
<p><strong>See also:</strong> <a href="#XREFlcm">lcm</a>, <a href="#XREFfactor">factor</a>, <a href="Predicates-for-Numeric-Objects.html#XREFisprime">isprime</a>.
</p></dd></dl>
<a name="XREFlcm"></a><dl>
<dt><a name="index-lcm"></a><em></em> <strong>lcm</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dt><a name="index-lcm-1"></a><em></em> <strong>lcm</strong> <em>(<var>x</var>, <var>y</var>, …)</em></dt>
<dd><p>Compute the least common multiple of <var>x</var> and <var>y</var>, or of the list of
all arguments.
</p>
<p>All elements must be numeric and of the same size or scalar.
</p>
<p><strong>See also:</strong> <a href="#XREFfactor">factor</a>, <a href="#XREFgcd">gcd</a>, <a href="Predicates-for-Numeric-Objects.html#XREFisprime">isprime</a>.
</p></dd></dl>
<a name="XREFrem"></a><dl>
<dt><a name="index-rem"></a><em></em> <strong>rem</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dd><p>Return the remainder of the division <code><var>x</var> / <var>y</var></code>.
</p>
<p>The remainder is computed using the expression
</p>
<div class="example">
<pre class="example">x - y .* fix (x ./ y)
</pre></div>
<p>An error message is printed if the dimensions of the arguments do not agree,
or if either argument is complex.
</p>
<p>Programming Notes: When calculating with floating point numbers (double,
single), values within a few eps of an integer will be rounded to that
integer before computation for compatibility with <small>MATLAB</small>. Any floating
point integers greater than <code>flintmax</code> (2^53 for double) will not compute
correctly. For larger integer values convert the input to <code>uint64</code> before
calling this function.
</p>
<p>By convention,
</p>
<div class="example">
<pre class="example">rem (<var>x</var>, 0) = NaN if <var>x</var> is a floating point variable
rem (<var>x</var>, 0) = 0 if <var>x</var> is an integer variable
rem (<var>x</var>, <var>y</var>) returns a value with the signbit from <var>x</var>
</pre></div>
<p>For the opposite conventions see the <code>mod</code> function. In general,
<code>rem</code> is best when computing the remainder after division of two
<em>positive</em> numbers. For negative numbers, or when the values are
periodic, <code>mod</code> is a better choice.
</p>
<p><strong>See also:</strong> <a href="#XREFmod">mod</a>.
</p></dd></dl>
<a name="XREFmod"></a><dl>
<dt><a name="index-mod"></a><em></em> <strong>mod</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dd><p>Compute the modulo of <var>x</var> and <var>y</var>.
</p>
<p>Conceptually this is given by
</p>
<div class="example">
<pre class="example">x - y .* floor (x ./ y)
</pre></div>
<p>and is written such that the correct modulus is returned for integer types.
This function handles negative values correctly. That is,
<code>mod <span class="nolinebreak">(-1,</span> 3)</code><!-- /@w --> is 2, not -1, as <code>rem <span class="nolinebreak">(-1,</span> 3)</code><!-- /@w --> returns.
</p>
<p>An error results if the dimensions of the arguments do not agree, or if
either of the arguments is complex.
</p>
<p>Programming Notes: When calculating with floating point numbers (double,
single), values within a few eps of an integer will be rounded to that
integer before computation for compatibility with <small>MATLAB</small>. Any floating
point integers greater than <code>flintmax</code> (2^53 for double) will not compute
correctly. For larger integer values convert the input to <code>uint64</code> before
calling this function.
</p>
<p>By convention,
</p>
<div class="example">
<pre class="example">mod (<var>x</var>, 0) = <var>x</var>
mod (<var>x</var>, <var>y</var>) returns a value with the signbit from <var>y</var>
</pre></div>
<p>For the opposite conventions see the <code>rem</code> function. In general,
<code>mod</code> is a better choice than <code>rem</code> when any of the inputs are
negative numbers or when the values are periodic.
</p>
<p><strong>See also:</strong> <a href="#XREFrem">rem</a>.
</p></dd></dl>
<a name="XREFprimes"></a><dl>
<dt><a name="index-primes"></a><em><var>p</var> =</em> <strong>primes</strong> <em>(<var>n</var>)</em></dt>
<dd><p>Return all primes up to <var>n</var>.
</p>
<p>The output data class (double, single, uint32, etc.) is the same as the
input class of <var>n</var>. The algorithm used is the Sieve of Eratosthenes.
</p>
<p>Note: If you need a specific number of primes you can use the fact that the
distance from one prime to the next is, on average, proportional to the
logarithm of the prime. Integrating, one finds that there are about
<em>k</em> primes less than
k*log (5*k).
</p>
<p>See also <code>list_primes</code> if you need a specific number <var>n</var> of primes.
</p>
<p><strong>See also:</strong> <a href="#XREFlist_005fprimes">list_primes</a>, <a href="Predicates-for-Numeric-Objects.html#XREFisprime">isprime</a>.
</p></dd></dl>
<a name="XREFlist_005fprimes"></a><dl>
<dt><a name="index-list_005fprimes"></a><em></em> <strong>list_primes</strong> <em>()</em></dt>
<dt><a name="index-list_005fprimes-1"></a><em></em> <strong>list_primes</strong> <em>(<var>n</var>)</em></dt>
<dd><p>List the first <var>n</var> primes.
</p>
<p>If <var>n</var> is unspecified, the first 25 primes are listed.
</p>
<p><strong>See also:</strong> <a href="#XREFprimes">primes</a>, <a href="Predicates-for-Numeric-Objects.html#XREFisprime">isprime</a>.
</p></dd></dl>
<a name="XREFsign"></a><dl>
<dt><a name="index-sign"></a><em></em> <strong>sign</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the <em>signum</em> function.
</p>
<p>This is defined as
</p>
<div class="example">
<pre class="example"> -1, x < 0;
sign (x) = 0, x = 0;
1, x > 0.
</pre></div>
<p>For complex arguments, <code>sign</code> returns <code>x ./ abs (<var>x</var>)</code>.
</p>
<p>Note that <code>sign (-0.0)</code> is 0. Although IEEE 754 floating point
allows zero to be signed, 0.0 and -0.0 compare equal. If you must test
whether zero is signed, use the <code>signbit</code> function.
</p>
<p><strong>See also:</strong> <a href="#XREFsignbit">signbit</a>.
</p></dd></dl>
<a name="XREFsignbit"></a><dl>
<dt><a name="index-signbit"></a><em></em> <strong>signbit</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Return logical true if the value of <var>x</var> has its sign bit set and false
otherwise.
</p>
<p>This behavior is consistent with the other logical functions.
See <a href="Logical-Values.html#Logical-Values">Logical Values</a>. The behavior differs from the C language function
which returns nonzero if the sign bit is set.
</p>
<p>This is not the same as <code>x < 0.0</code>, because IEEE 754 floating point
allows zero to be signed. The comparison <code>-0.0 < 0.0</code> is false,
but <code>signbit (-0.0)</code> will return a nonzero value.
</p>
<p><strong>See also:</strong> <a href="#XREFsign">sign</a>.
</p></dd></dl>
<hr>
<div class="header">
<p>
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