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<html lang="en">
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<p>
<a name="Utility-Functions"></a>
Next: <a rel="next" accesskey="n" href="Special-Functions.html#Special-Functions">Special Functions</a>,
Previous: <a rel="previous" accesskey="p" href="Sums-and-Products.html#Sums-and-Products">Sums and Products</a>,
Up: <a rel="up" accesskey="u" href="Arithmetic.html#Arithmetic">Arithmetic</a>
<hr>
</div>
<h3 class="section">17.5 Utility Functions</h3>
<!-- mappers.cc -->
<p><a name="doc_002dceil"></a>
<div class="defun">
— Mapping Function: <b>ceil</b> (<var>x</var>)<var><a name="index-ceil-1436"></a></var><br>
<blockquote><p>Return the smallest integer not less than <var>x</var>. This is equivalent to
rounding towards positive infinity. If <var>x</var> is
complex, return <code>ceil (real (</code><var>x</var><code>)) + ceil (imag (</code><var>x</var><code>)) * I</code>.
<pre class="example"> ceil ([-2.7, 2.7])
-2 3
</pre>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dfloor.html#doc_002dfloor">floor</a>, <a href="doc_002dround.html#doc_002dround">round</a>, <a href="doc_002dfix.html#doc_002dfix">fix</a>.
</p></blockquote></div>
<!-- ./linear-algebra/cross.m -->
<p><a name="doc_002dcross"></a>
<div class="defun">
— Function File: <b>cross</b> (<var>x, y</var>)<var><a name="index-cross-1437"></a></var><br>
— Function File: <b>cross</b> (<var>x, y, dim</var>)<var><a name="index-cross-1438"></a></var><br>
<blockquote><p>Compute the vector cross product of two 3-dimensional vectors
<var>x</var> and <var>y</var>.
<pre class="example"> cross ([1,1,0], [0,1,1])
[ 1; -1; 1 ]
</pre>
<p>If <var>x</var> and <var>y</var> are matrices, the cross product is applied
along the first dimension with 3 elements. The optional argument
<var>dim</var> forces the cross product to be calculated along
the specified dimension.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002ddot.html#doc_002ddot">dot</a>.
</p></blockquote></div>
<!-- ./general/del2.m -->
<p><a name="doc_002ddel2"></a>
<div class="defun">
— Function File: <var>d</var> = <b>del2</b> (<var>m</var>)<var><a name="index-del2-1439"></a></var><br>
— Function File: <var>d</var> = <b>del2</b> (<var>m, h</var>)<var><a name="index-del2-1440"></a></var><br>
— Function File: <var>d</var> = <b>del2</b> (<var>m, dx, dy, <small class="dots">...</small></var>)<var><a name="index-del2-1441"></a></var><br>
<blockquote>
<p>Calculate the discrete Laplace
operator.
For a 2-dimensional matrix <var>m</var> this is defined as
<pre class="example"> 1 / d^2 d^2 \
D = --- * | --- M(x,y) + --- M(x,y) |
4 \ dx^2 dy^2 /
</pre>
<p>For N-dimensional arrays the sum in parentheses is expanded to include second derivatives
over the additional higher dimensions.
<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>At least 3 data points are needed for each dimension. Boundary points are
calculated from the linear extrapolation of interior points.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dgradient.html#doc_002dgradient">gradient</a>, <a href="doc_002ddiff.html#doc_002ddiff">diff</a>.
</p></blockquote></div>
<!-- ./specfun/factor.m -->
<p><a name="doc_002dfactor"></a>
<div class="defun">
— Function File: <var>p</var> = <b>factor</b> (<var>q</var>)<var><a name="index-factor-1442"></a></var><br>
— Function File: [<var>p</var>, <var>n</var>] = <b>factor</b> (<var>q</var>)<var><a name="index-factor-1443"></a></var><br>
<blockquote>
<p>Return prime factorization of <var>q</var>. That is, <code>prod (</code><var>p</var><code>)
== </code><var>q</var> and every element of <var>p</var> is a prime number. If
<var>q</var><code> == 1</code>, returns 1.
<p>With two output arguments, return the unique primes <var>p</var> and
their multiplicities. That is, <code>prod (</code><var>p</var><code> .^ </code><var>n</var><code>) ==
</code><var>q</var>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dgcd.html#doc_002dgcd">gcd</a>, <a href="doc_002dlcm.html#doc_002dlcm">lcm</a>.
</p></blockquote></div>
<!-- ./specfun/factorial.m -->
<p><a name="doc_002dfactorial"></a>
<div class="defun">
— Function File: <b>factorial</b> (<var>n</var>)<var><a name="index-factorial-1444"></a></var><br>
<blockquote><p>Return the factorial of <var>n</var> where <var>n</var> is a positive integer. If
<var>n</var> is a scalar, this is equivalent to <code>prod (1:</code><var>n</var><code>)</code>. For
vector or matrix arguments, return the factorial of each element in the
array. For non-integers see the generalized factorial function
<code>gamma</code>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dprod.html#doc_002dprod">prod</a>, <a href="doc_002dgamma.html#doc_002dgamma">gamma</a>.
</p></blockquote></div>
<!-- mappers.cc -->
<p><a name="doc_002dfix"></a>
<div class="defun">
— Mapping Function: <b>fix</b> (<var>x</var>)<var><a name="index-fix-1445"></a></var><br>
<blockquote><p>Truncate fractional portion of <var>x</var> and return the integer portion. This
is equivalent to rounding towards zero. If <var>x</var> is complex, return
<code>fix (real (</code><var>x</var><code>)) + fix (imag (</code><var>x</var><code>)) * I</code>.
<pre class="example"> fix ([-2.7, 2.7])
-2 2
</pre>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dceil.html#doc_002dceil">ceil</a>, <a href="doc_002dfloor.html#doc_002dfloor">floor</a>, <a href="doc_002dround.html#doc_002dround">round</a>.
</p></blockquote></div>
<!-- mappers.cc -->
<p><a name="doc_002dfloor"></a>
<div class="defun">
— Mapping Function: <b>floor</b> (<var>x</var>)<var><a name="index-floor-1446"></a></var><br>
<blockquote><p>Return the largest integer not greater than <var>x</var>. This is equivalent to
rounding towards negative infinity. If <var>x</var> is
complex, return <code>floor (real (</code><var>x</var><code>)) + floor (imag (</code><var>x</var><code>)) * I</code>.
<pre class="example"> floor ([-2.7, 2.7])
-3 2
</pre>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dceil.html#doc_002dceil">ceil</a>, <a href="doc_002dround.html#doc_002dround">round</a>, <a href="doc_002dfix.html#doc_002dfix">fix</a>.
</p></blockquote></div>
<!-- data.cc -->
<p><a name="doc_002dfmod"></a>
<div class="defun">
— Mapping Function: <b>fmod</b> (<var>x, y</var>)<var><a name="index-fmod-1447"></a></var><br>
<blockquote><p>Compute the floating point remainder of dividing <var>x</var> by <var>y</var>
using the C library function <code>fmod</code>. The result has the same
sign as <var>x</var>. If <var>y</var> is zero, the result is implementation-dependent.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dmod.html#doc_002dmod">mod</a>, <a href="doc_002drem.html#doc_002drem">rem</a>.
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/gcd.cc -->
<p><a name="doc_002dgcd"></a>
<div class="defun">
— Loadable Function: <var>g</var> = <b>gcd</b> (<var>a</var>)<var><a name="index-gcd-1448"></a></var><br>
— Loadable Function: <var>g</var> = <b>gcd</b> (<var>a1, a2, <small class="dots">...</small></var>)<var><a name="index-gcd-1449"></a></var><br>
— Loadable Function: [<var>g</var>, <var>v1</var>, <small class="dots">...</small>] = <b>gcd</b> (<var>a1, a2, <small class="dots">...</small></var>)<var><a name="index-gcd-1450"></a></var><br>
<blockquote>
<p>Compute the greatest common divisor of the elements of <var>a</var>. If more
than one argument is given 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 integers. For example,
<pre class="example"> gcd ([15, 20])
5
</pre>
<p class="noindent">and
<pre class="example"> gcd ([15, 9], [20, 18])
5 9
</pre>
<p>Optional return arguments <var>v1</var>, etc., contain integer vectors such
that,
<pre class="example"> <var>g</var> = <var>v1</var> .* <var>a1</var> + <var>v2</var> .* <var>a2</var> + ...
</pre>
<p>For backward compatibility with previous versions of this function, when
all arguments are scalar, a single return argument <var>v1</var> containing
all of the values of <var>v1</var>, <small class="dots">...</small> is acceptable.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dlcm.html#doc_002dlcm">lcm</a>, <a href="doc_002dfactor.html#doc_002dfactor">factor</a>.
</p></blockquote></div>
<!-- ./general/gradient.m -->
<p><a name="doc_002dgradient"></a>
<div class="defun">
— Function File: <var>dx</var> = <b>gradient</b> (<var>m</var>)<var><a name="index-gradient-1451"></a></var><br>
— Function File: [<var>dx</var>, <var>dy</var>, <var>dz</var>, <small class="dots">...</small>] = <b>gradient</b> (<var>m</var>)<var><a name="index-gradient-1452"></a></var><br>
— Function File: [<small class="dots">...</small>] = <b>gradient</b> (<var>m, s</var>)<var><a name="index-gradient-1453"></a></var><br>
— Function File: [<small class="dots">...</small>] = <b>gradient</b> (<var>m, x, y, z, <small class="dots">...</small></var>)<var><a name="index-gradient-1454"></a></var><br>
— Function File: [<small class="dots">...</small>] = <b>gradient</b> (<var>f, x0</var>)<var><a name="index-gradient-1455"></a></var><br>
— Function File: [<small class="dots">...</small>] = <b>gradient</b> (<var>f, x0, s</var>)<var><a name="index-gradient-1456"></a></var><br>
— Function File: [<small class="dots">...</small>] = <b>gradient</b> (<var>f, x0, x, y, <small class="dots">...</small></var>)<var><a name="index-gradient-1457"></a></var><br>
<blockquote>
<p>Calculate the gradient of sampled data or a function. 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><code>[</code><var>dx</var><code>, </code><var>dy</var><code>] = gradient (</code><var>m</var><code>)</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>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>, <small class="dots">...</small> arguments. Scalar values specify an equidistant spacing.
Vector values for the <var>x</var>, <small class="dots">...</small> arguments specify the coordinate for that
dimension. The length must match their respective dimension of <var>m</var>.
<p>At boundary points a linear extrapolation is applied. Interior points
are calculated with the first approximation of the numerical gradient
<pre class="example"> y'(i) = 1/(x(i+1)-x(i-1)) * (y(i-1)-y(i+1)).
</pre>
<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 x0 = 0. 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>, <small class="dots">...</small> arguments. By default a spacing of 1 is used.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002ddiff.html#doc_002ddiff">diff</a>, <a href="doc_002ddel2.html#doc_002ddel2">del2</a>.
</p></blockquote></div>
<!-- data.cc -->
<p><a name="doc_002dhypot"></a>
<div class="defun">
— Built-in Function: <b>hypot</b> (<var>x, y</var>)<var><a name="index-hypot-1458"></a></var><br>
<blockquote><p>Compute the element-by-element square root of the sum of the squares of
<var>x</var> and <var>y</var>. This is equivalent to
<code>sqrt (</code><var>x</var><code>.^2 + </code><var>y</var><code>.^2)</code>, but calculated in a manner that
avoids overflows for large values of <var>x</var> or <var>y</var>.
</p></blockquote></div>
<!-- ./elfun/lcm.m -->
<p><a name="doc_002dlcm"></a>
<div class="defun">
— Mapping Function: <b>lcm</b> (<var>x</var>)<var><a name="index-lcm-1459"></a></var><br>
— Mapping Function: <b>lcm</b> (<var>x, <small class="dots">...</small></var>)<var><a name="index-lcm-1460"></a></var><br>
<blockquote><p>Compute the least common multiple of the elements of <var>x</var>, or
of the list of all arguments. For example,
<pre class="example"> lcm (a1, ..., ak)
</pre>
<p class="noindent">is the same as
<pre class="example"> lcm ([a1, ..., ak]).
</pre>
<p>All elements must be the same size or scalar.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dfactor.html#doc_002dfactor">factor</a>, <a href="doc_002dgcd.html#doc_002dgcd">gcd</a>.
</p></blockquote></div>
<!-- ./miscellaneous/list_primes.m -->
<p><a name="doc_002dlist_005fprimes"></a>
<div class="defun">
— Function File: <b>list_primes</b> (<var>n</var>)<var><a name="index-list_005fprimes-1461"></a></var><br>
<blockquote><p>List the first <var>n</var> primes. If <var>n</var> is unspecified, the first
25 primes are listed.
<p>The algorithm used is from page 218 of the TeXbook.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dprimes.html#doc_002dprimes">primes</a>, <a href="doc_002disprime.html#doc_002disprime">isprime</a>.
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/max.cc -->
<p><a name="doc_002dmax"></a>
<div class="defun">
— Loadable Function: <b>max</b> (<var>x</var>)<var><a name="index-max-1462"></a></var><br>
— Loadable Function: <b>max</b> (<var>x, y</var>)<var><a name="index-max-1463"></a></var><br>
— Loadable Function: <b>max</b> (<var>x, y, dim</var>)<var><a name="index-max-1464"></a></var><br>
— Loadable Function: [<var>w</var>, <var>iw</var>] = <b>max</b> (<var>x</var>)<var><a name="index-max-1465"></a></var><br>
<blockquote><p>For a vector argument, return the maximum value. For a matrix
argument, return the maximum value from each column, as a row
vector, or over the dimension <var>dim</var> if defined. For two matrices
(or a matrix and scalar), return the pair-wise maximum.
Thus,
<pre class="example"> max (max (<var>x</var>))
</pre>
<p class="noindent">returns the largest element of the matrix <var>x</var>, and
<pre class="example"> max (2:5, pi)
3.1416 3.1416 4.0000 5.0000
</pre>
<p class="noindent">compares each element of the range <code>2:5</code> with <code>pi</code>, and
returns a row vector of the maximum values.
<p>For complex arguments, the magnitude of the elements are used for
comparison.
<p>If called with one input and two output arguments,
<code>max</code> also returns the first index of the
maximum value(s). Thus,
<pre class="example"> [x, ix] = max ([1, 3, 5, 2, 5])
x = 5
ix = 3
</pre>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dmin.html#doc_002dmin">min</a>, <a href="doc_002dcummax.html#doc_002dcummax">cummax</a>, <a href="doc_002dcummin.html#doc_002dcummin">cummin</a>.
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/max.cc -->
<p><a name="doc_002dmin"></a>
<div class="defun">
— Loadable Function: <b>min</b> (<var>x</var>)<var><a name="index-min-1466"></a></var><br>
— Loadable Function: <b>min</b> (<var>x, y</var>)<var><a name="index-min-1467"></a></var><br>
— Loadable Function: <b>min</b> (<var>x, y, dim</var>)<var><a name="index-min-1468"></a></var><br>
— Loadable Function: [<var>w</var>, <var>iw</var>] = <b>min</b> (<var>x</var>)<var><a name="index-min-1469"></a></var><br>
<blockquote><p>For a vector argument, return the minimum value. For a matrix
argument, return the minimum value from each column, as a row
vector, or over the dimension <var>dim</var> if defined. For two matrices
(or a matrix and scalar), return the pair-wise minimum.
Thus,
<pre class="example"> min (min (<var>x</var>))
</pre>
<p class="noindent">returns the smallest element of <var>x</var>, and
<pre class="example"> min (2:5, pi)
2.0000 3.0000 3.1416 3.1416
</pre>
<p class="noindent">compares each element of the range <code>2:5</code> with <code>pi</code>, and
returns a row vector of the minimum values.
<p>For complex arguments, the magnitude of the elements are used for
comparison.
<p>If called with one input and two output arguments,
<code>min</code> also returns the first index of the
minimum value(s). Thus,
<pre class="example"> [x, ix] = min ([1, 3, 0, 2, 0])
x = 0
ix = 3
</pre>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dmax.html#doc_002dmax">max</a>, <a href="doc_002dcummin.html#doc_002dcummin">cummin</a>, <a href="doc_002dcummax.html#doc_002dcummax">cummax</a>.
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/max.cc -->
<p><a name="doc_002dcummax"></a>
<div class="defun">
— Loadable Function: <b>cummax</b> (<var>x</var>)<var><a name="index-cummax-1470"></a></var><br>
— Loadable Function: <b>cummax</b> (<var>x, dim</var>)<var><a name="index-cummax-1471"></a></var><br>
— Loadable Function: [<var>w</var>, <var>iw</var>] = <b>cummax</b> (<var>x</var>)<var><a name="index-cummax-1472"></a></var><br>
<blockquote><p>Return the cumulative maximum values along dimension <var>dim</var>. If <var>dim</var>
is unspecified it defaults to column-wise operation. For example,
<pre class="example"> cummax ([1 3 2 6 4 5])
1 3 3 6 6 6
</pre>
<p>The call
<pre class="example"> [w, iw] = cummax (x, dim)
</pre>
<p class="noindent">is equivalent to the following code:
<pre class="example"> w = iw = zeros (size (x));
idxw = idxx = repmat ({':'}, 1, ndims (x));
for i = 1:size (x, dim)
idxw{dim} = i; idxx{dim} = 1:i;
[w(idxw{:}), iw(idxw{:})] = max(x(idxx{:}), [], dim);
endfor
</pre>
<p class="noindent">but computed in a much faster manner.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dcummin.html#doc_002dcummin">cummin</a>, <a href="doc_002dmax.html#doc_002dmax">max</a>, <a href="doc_002dmin.html#doc_002dmin">min</a>.
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/max.cc -->
<p><a name="doc_002dcummin"></a>
<div class="defun">
— Loadable Function: <b>cummin</b> (<var>x</var>)<var><a name="index-cummin-1473"></a></var><br>
— Loadable Function: <b>cummin</b> (<var>x, dim</var>)<var><a name="index-cummin-1474"></a></var><br>
— Loadable Function: [<var>w</var>, <var>iw</var>] = <b>cummin</b> (<var>x</var>)<var><a name="index-cummin-1475"></a></var><br>
<blockquote><p>Return the cumulative minimum values along dimension <var>dim</var>. If <var>dim</var>
is unspecified it defaults to column-wise operation. For example,
<pre class="example"> cummin ([5 4 6 2 3 1])
5 4 4 2 2 1
</pre>
<p>The call
<pre class="example"> [w, iw] = cummin (x, dim)
</pre>
<p class="noindent">is equivalent to the following code:
<pre class="example"> w = iw = zeros (size (x));
idxw = idxx = repmat ({':'}, 1, ndims (x));
for i = 1:size (x, dim)
idxw{dim} = i; idxx{dim} = 1:i;
[w(idxw{:}), iw(idxw{:})] = min(x(idxx{:}), [], dim);
endfor
</pre>
<p class="noindent">but computed in a much faster manner.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dcummax.html#doc_002dcummax">cummax</a>, <a href="doc_002dmin.html#doc_002dmin">min</a>, <a href="doc_002dmax.html#doc_002dmax">max</a>.
</p></blockquote></div>
<!-- ./general/mod.m -->
<p><a name="doc_002dmod"></a>
<div class="defun">
— Mapping Function: <b>mod</b> (<var>x, y</var>)<var><a name="index-mod-1476"></a></var><br>
<blockquote><p>Compute the modulo of <var>x</var> and <var>y</var>. Conceptually this is given by
<pre class="example"> x - y .* floor (x ./ y)
</pre>
<p>and is written such that the correct modulus is returned for
integer types. This function handles negative values correctly. That
is, <code>mod (-1, 3)</code> is 2, not -1, as <code>rem (-1, 3)</code> returns.
<code>mod (</code><var>x</var><code>, 0)</code> returns <var>x</var>.
<p>An error results if the dimensions of the arguments do not agree, or if
either of the arguments is complex.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002drem.html#doc_002drem">rem</a>, <a href="doc_002dfmod.html#doc_002dfmod">fmod</a>.
</p></blockquote></div>
<!-- ./specfun/primes.m -->
<p><a name="doc_002dprimes"></a>
<div class="defun">
— Function File: <b>primes</b> (<var>n</var>)<var><a name="index-primes-1477"></a></var><br>
<blockquote>
<p>Return all primes up to <var>n</var>.
<p>The algorithm used is the Sieve of Erastothenes.
<p>Note that if you need a specific number of primes you can use the
fact 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 k primes less than
k*log(5*k).
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dlist_005fprimes.html#doc_002dlist_005fprimes">list_primes</a>, <a href="doc_002disprime.html#doc_002disprime">isprime</a>.
</p></blockquote></div>
<!-- ./general/rem.m -->
<p><a name="doc_002drem"></a>
<div class="defun">
— Mapping Function: <b>rem</b> (<var>x, y</var>)<var><a name="index-rem-1478"></a></var><br>
<blockquote><p>Return the remainder of the division <var>x</var><code> / </code><var>y</var>, computed
using the expression
<pre class="example"> x - y .* fix (x ./ y)
</pre>
<p>An error message is printed if the dimensions of the arguments do not
agree, or if either of the arguments is complex.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dmod.html#doc_002dmod">mod</a>, <a href="doc_002dfmod.html#doc_002dfmod">fmod</a>.
</p></blockquote></div>
<!-- mappers.cc -->
<p><a name="doc_002dround"></a>
<div class="defun">
— Mapping Function: <b>round</b> (<var>x</var>)<var><a name="index-round-1479"></a></var><br>
<blockquote><p>Return the integer nearest to <var>x</var>. If <var>x</var> is complex, return
<code>round (real (</code><var>x</var><code>)) + round (imag (</code><var>x</var><code>)) * I</code>.
<pre class="example"> round ([-2.7, 2.7])
-3 3
</pre>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dceil.html#doc_002dceil">ceil</a>, <a href="doc_002dfloor.html#doc_002dfloor">floor</a>, <a href="doc_002dfix.html#doc_002dfix">fix</a>.
</p></blockquote></div>
<!-- mappers.cc -->
<p><a name="doc_002droundb"></a>
<div class="defun">
— Mapping Function: <b>roundb</b> (<var>x</var>)<var><a name="index-roundb-1480"></a></var><br>
<blockquote><p>Return the integer nearest to <var>x</var>. If there are two nearest
integers, return the even one (banker's rounding). If <var>x</var> is complex,
return <code>roundb (real (</code><var>x</var><code>)) + roundb (imag (</code><var>x</var><code>)) * I</code>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dround.html#doc_002dround">round</a>.
</p></blockquote></div>
<!-- mappers.cc -->
<p><a name="doc_002dsign"></a>
<div class="defun">
— Mapping Function: <b>sign</b> (<var>x</var>)<var><a name="index-sign-1481"></a></var><br>
<blockquote><p>Compute the <dfn>signum</dfn> function, which is defined as
<pre class="example"> -1, x < 0;
sign (x) = 0, x = 0;
1, x > 0.
</pre>
<p>For complex arguments, <code>sign</code> returns <code>x ./ abs (</code><var>x</var><code>)</code>.
</p></blockquote></div>
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