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<a name="Functions-of-One-Variable"></a>
<p>
Next: <a rel="next" accesskey="n" href="Functions-of-Multiple-Variables.html#Functions-of-Multiple-Variables">Functions of Multiple Variables</a>,
Up: <a rel="up" accesskey="u" href="Numerical-Integration.html#Numerical-Integration">Numerical Integration</a>
<hr>
</div>
<h3 class="section">22.1 Functions of One Variable</h3>
<p>Octave supports three different algorithms for computing the integral
of a function f over the interval from a to b.
These are
<dl>
<dt><code>quad</code><dd>Numerical integration based on Gaussian quadrature.
<br><dt><code>quadl</code><dd>Numerical integration using an adaptive Lobatto rule.
<br><dt><code>quadgk</code><dd>Numerical integration using an adaptive Gauss-Konrod rule.
<br><dt><code>quadv</code><dd>Numerical integration using an adaptive vectorized Simpson's rule.
<br><dt><code>trapz</code><dd>Numerical integration using the trapezoidal method.
</dl>
<p class="noindent">Besides these functions Octave also allows you to perform cumulative
numerical integration using the trapezoidal method through the
<code>cumtrapz</code> function.
<!-- ./DLD-FUNCTIONS/quad.cc -->
<p><a name="doc_002dquad"></a>
<div class="defun">
— Loadable Function: [<var>v</var>, <var>ier</var>, <var>nfun</var>, <var>err</var>] = <b>quad</b> (<var>f, a, b, tol, sing</var>)<var><a name="index-quad-1760"></a></var><br>
<blockquote><p>Integrate a nonlinear function of one variable using Quadpack.
The first argument is the name of the function, the function handle or
the inline function to call to compute the value of the integrand. It
must have the form
<pre class="example"> y = f (x)
</pre>
<p class="noindent">where <var>y</var> and <var>x</var> are scalars.
<p>The second and third arguments are limits of integration. Either or
both may be infinite.
<p>The optional argument <var>tol</var> is a vector that specifies the desired
accuracy of the result. The first element of the vector is the desired
absolute tolerance, and the second element is the desired relative
tolerance. To choose a relative test only, set the absolute
tolerance to zero. To choose an absolute test only, set the relative
tolerance to zero.
<p>The optional argument <var>sing</var> is a vector of values at which the
integrand is known to be singular.
<p>The result of the integration is returned in <var>v</var> and <var>ier</var>
contains an integer error code (0 indicates a successful integration).
The value of <var>nfun</var> indicates how many function evaluations were
required, and <var>err</var> contains an estimate of the error in the
solution.
<p>You can use the function <code>quad_options</code> to set optional
parameters for <code>quad</code>.
<p>It should be noted that since <code>quad</code> is written in Fortran it
cannot be called recursively.
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/quad.cc -->
<p><a name="doc_002dquad_005foptions"></a>
<div class="defun">
— Loadable Function: <b>quad_options</b> (<var>opt, val</var>)<var><a name="index-quad_005foptions-1761"></a></var><br>
<blockquote><p>When called with two arguments, this function
allows you set options parameters for the function <code>quad</code>.
Given one argument, <code>quad_options</code> returns the value of the
corresponding option. If no arguments are supplied, the names of all
the available options and their current values are displayed.
<p>Options include
<dl>
<dt><code>"absolute tolerance"</code><dd>Absolute tolerance; may be zero for pure relative error test.
<br><dt><code>"relative tolerance"</code><dd>Nonnegative relative tolerance. If the absolute tolerance is zero,
the relative tolerance must be greater than or equal to
<code>max (50*eps, 0.5e-28)</code>.
<br><dt><code>"single precision absolute tolerance"</code><dd>Absolute tolerance for single precision; may be zero for pure relative
error test.
<br><dt><code>"single precision relative tolerance"</code><dd>Nonnegative relative tolerance for single precision. If the absolute
tolerance is zero, the relative tolerance must be greater than or equal to
<code>max (50*eps, 0.5e-28)</code>.
</dl>
</p></blockquote></div>
<p>Here is an example of using <code>quad</code> to integrate the function
<pre class="example"> <var>f</var>(<var>x</var>) = <var>x</var> * sin (1/<var>x</var>) * sqrt (abs (1 - <var>x</var>))
</pre>
<p class="noindent">from <var>x</var> = 0 to <var>x</var> = 3.
<p>This is a fairly difficult integration (plot the function over the range
of integration to see why).
<p>The first step is to define the function:
<pre class="example"> function y = f (x)
y = x .* sin (1 ./ x) .* sqrt (abs (1 - x));
endfunction
</pre>
<p>Note the use of the `dot' forms of the operators. This is not necessary
for the call to <code>quad</code>, but it makes it much easier to generate a
set of points for plotting (because it makes it possible to call the
function with a vector argument to produce a vector result).
<p>Then we simply call quad:
<pre class="example"> [v, ier, nfun, err] = quad ("f", 0, 3)
⇒ 1.9819
⇒ 1
⇒ 5061
⇒ 1.1522e-07
</pre>
<p>Although <code>quad</code> returns a nonzero value for <var>ier</var>, the result
is reasonably accurate (to see why, examine what happens to the result
if you move the lower bound to 0.1, then 0.01, then 0.001, etc.).
<!-- ./general/quadl.m -->
<p><a name="doc_002dquadl"></a>
<div class="defun">
— Function File: <var>q</var> = <b>quadl</b> (<var>f, a, b</var>)<var><a name="index-quadl-1762"></a></var><br>
— Function File: <var>q</var> = <b>quadl</b> (<var>f, a, b, tol</var>)<var><a name="index-quadl-1763"></a></var><br>
— Function File: <var>q</var> = <b>quadl</b> (<var>f, a, b, tol, trace</var>)<var><a name="index-quadl-1764"></a></var><br>
— Function File: <var>q</var> = <b>quadl</b> (<var>f, a, b, tol, trace, p1, p2, <small class="dots">...</small></var>)<var><a name="index-quadl-1765"></a></var><br>
<blockquote>
<p>Numerically evaluate integral using adaptive Lobatto rule.
<code>quadl (</code><var>f</var><code>, </code><var>a</var><code>, </code><var>b</var><code>)</code> approximates the integral of
<var>f</var><code>(</code><var>x</var><code>)</code> to machine precision. <var>f</var> is either a
function handle, inline function or string containing the name of
the function to evaluate. The function <var>f</var> must return a vector
of output values if given a vector of input values.
<p>If defined, <var>tol</var> defines the relative tolerance to which to
which to integrate <var>f</var><code>(</code><var>x</var><code>)</code>. While if <var>trace</var> is
defined, displays the left end point of the current interval, the
interval length, and the partial integral.
<p>Additional arguments <var>p1</var>, etc., are passed directly to <var>f</var>.
To use default values for <var>tol</var> and <var>trace</var>, one may pass
empty matrices.
<p>Reference: W. Gander and W. Gautschi, 'Adaptive Quadrature -
Revisited', BIT Vol. 40, No. 1, March 2000, pp. 84–101.
<a href="http://www.inf.ethz.ch/personal/gander/">http://www.inf.ethz.ch/personal/gander/</a>
</blockquote></div>
<!-- ./general/quadgk.m -->
<p><a name="doc_002dquadgk"></a>
<div class="defun">
— Function File: <b>quadgk</b> (<var>f, a, b, abstol, trace</var>)<var><a name="index-quadgk-1766"></a></var><br>
— Function File: <b>quadgk</b> (<var>f, a, b, prop, val, <small class="dots">...</small></var>)<var><a name="index-quadgk-1767"></a></var><br>
— Function File: [<var>q</var>, <var>err</var>] = <b>quadgk</b> (<var><small class="dots">...</small></var>)<var><a name="index-quadgk-1768"></a></var><br>
<blockquote><p>Numerically evaluate integral using adaptive Gauss-Konrod quadrature.
The formulation is based on a proposal by L.F. Shampine,
<cite>"Vectorized adaptive quadrature in </cite><span class="sc">matlab</span><cite>", Journal of
Computational and Applied Mathematics, pp131-140, Vol 211, Issue 2,
Feb 2008</cite> where all function evaluations at an iteration are
calculated with a single call to <var>f</var>. Therefore the function
<var>f</var> must be of the form <var>f</var><code> (</code><var>x</var><code>)</code> and accept
vector values of <var>x</var> and return a vector of the same length
representing the function evaluations at the given values of <var>x</var>.
The function <var>f</var> can be defined in terms of a function handle,
inline function or string.
<p>The bounds of the quadrature <code>[</code><var>a</var><code>, </code><var>b</var><code>]</code> can be finite
or infinite and contain weak end singularities. Variable
transformation will be used to treat infinite intervals and weaken
the singularities. For example
<pre class="example"> quadgk(@(x) 1 ./ (sqrt (x) .* (x + 1)), 0, Inf)
</pre>
<p class="noindent">Note that the formulation of the integrand uses the
element-by-element operator <code>./</code> and all user functions to
<code>quadgk</code> should do the same.
<p>The absolute tolerance can be passed as a fourth argument in a manner
compatible with <code>quadv</code>. Equally the user can request that
information on the convergence can be printed is the fifth argument
is logically true.
<p>Alternatively, certain properties of <code>quadgk</code> can be passed as
pairs <var>prop</var><code>, </code><var>val</var>. Valid properties are
<dl>
<dt><code>AbsTol</code><dd>Defines the absolute error tolerance for the quadrature. The default
absolute tolerance is 1e-10.
<br><dt><code>RelTol</code><dd>Defines the relative error tolerance for the quadrature. The default
relative tolerance is 1e-5.
<br><dt><code>MaxIntervalCount</code><dd><code>quadgk</code> initially subdivides the interval on which to perform
the quadrature into 10 intervals. Sub-intervals that have an
unacceptable error are sub-divided and re-evaluated. If the number of
sub-intervals exceeds at any point 650 sub-intervals then a poor
convergence is signaled and the current estimate of the integral is
returned. The property 'MaxIntervalCount' can be used to alter the
number of sub-intervals that can exist before exiting.
<br><dt><code>WayPoints</code><dd>If there exists discontinuities in the first derivative of the
function to integrate, then these can be flagged with the
<code>"WayPoints"</code> property. This forces the ends of a sub-interval
to fall on the breakpoints of the function and can result in
significantly improved estimation of the error in the integral, faster
computation or both. For example,
<pre class="example"> quadgk (@(x) abs (1 - x .^ 2), 0, 2, 'Waypoints', 1)
</pre>
<p class="noindent">signals the breakpoint in the integrand at <var>x</var><code> = 1</code>.
<br><dt><code>Trace</code><dd>If logically true, then <code>quadgk</code> prints information on the
convergence of the quadrature at each iteration.
</dl>
<p>If any of <var>a</var>, <var>b</var> or <var>waypoints</var> is complex, then the
quadrature is treated as a contour integral along a piecewise
continuous path defined by the above. In this case the integral is
assumed to have no edge singularities. For example
<pre class="example"> quadgk (@(z) log (z), 1+1i, 1+1i, "WayPoints",
[1-1i, -1,-1i, -1+1i])
</pre>
<p class="noindent">integrates <code>log (z)</code> along the square defined by <code>[1+1i,
1-1i, -1-1i, -1+1i]</code>
<p>If two output arguments are requested, then <var>err</var> returns the
approximate bounds on the error in the integral <code>abs (</code><var>q</var><code> -
</code><var>i</var><code>)</code>, where <var>i</var> is the exact value of the integral.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dtriplequad.html#doc_002dtriplequad">triplequad</a>, <a href="doc_002ddblquad.html#doc_002ddblquad">dblquad</a>, <a href="doc_002dquad.html#doc_002dquad">quad</a>, <a href="doc_002dquadl.html#doc_002dquadl">quadl</a>, <a href="doc_002dquadv.html#doc_002dquadv">quadv</a>, <a href="doc_002dtrapz.html#doc_002dtrapz">trapz</a>.
</p></blockquote></div>
<!-- ./general/quadv.m -->
<p><a name="doc_002dquadv"></a>
<div class="defun">
— Function File: <var>q</var> = <b>quadv</b> (<var>f, a, b</var>)<var><a name="index-quadv-1769"></a></var><br>
— Function File: <var>q</var> = <b>quadl</b> (<var>f, a, b, tol</var>)<var><a name="index-quadl-1770"></a></var><br>
— Function File: <var>q</var> = <b>quadl</b> (<var>f, a, b, tol, trace</var>)<var><a name="index-quadl-1771"></a></var><br>
— Function File: <var>q</var> = <b>quadl</b> (<var>f, a, b, tol, trace, p1, p2, <small class="dots">...</small></var>)<var><a name="index-quadl-1772"></a></var><br>
— Function File: [<var>q</var>, <var>fcnt</var>] = <b>quadl</b> (<var><small class="dots">...</small></var>)<var><a name="index-quadl-1773"></a></var><br>
<blockquote>
<p>Numerically evaluate integral using adaptive Simpson's rule.
<code>quadv (</code><var>f</var><code>, </code><var>a</var><code>, </code><var>b</var><code>)</code> approximates the integral of
<var>f</var><code>(</code><var>x</var><code>)</code> to the default absolute tolerance of <code>1e-6</code>.
<var>f</var> is either a function handle, inline function or string
containing the name of the function to evaluate. The function <var>f</var>
must accept a string, and can return a vector representing the
approximation to <var>n</var> different sub-functions.
<p>If defined, <var>tol</var> defines the absolute tolerance to which to
which to integrate each sub-interval of <var>f</var><code>(</code><var>x</var><code>)</code>.
While if <var>trace</var> is defined, displays the left end point of the
current interval, the interval length, and the partial integral.
<p>Additional arguments <var>p1</var>, etc., are passed directly to <var>f</var>.
To use default values for <var>tol</var> and <var>trace</var>, one may pass
empty matrices.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dtriplequad.html#doc_002dtriplequad">triplequad</a>, <a href="doc_002ddblquad.html#doc_002ddblquad">dblquad</a>, <a href="doc_002dquad.html#doc_002dquad">quad</a>, <a href="doc_002dquadl.html#doc_002dquadl">quadl</a>, <a href="doc_002dquadgk.html#doc_002dquadgk">quadgk</a>, <a href="doc_002dtrapz.html#doc_002dtrapz">trapz</a>.
</p></blockquote></div>
<!-- ./general/trapz.m -->
<p><a name="doc_002dtrapz"></a>
<div class="defun">
— Function File: <var>z</var> = <b>trapz</b> (<var>y</var>)<var><a name="index-trapz-1774"></a></var><br>
— Function File: <var>z</var> = <b>trapz</b> (<var>x, y</var>)<var><a name="index-trapz-1775"></a></var><br>
— Function File: <var>z</var> = <b>trapz</b> (<var><small class="dots">...</small>, dim</var>)<var><a name="index-trapz-1776"></a></var><br>
<blockquote>
<p>Numerical integration using trapezoidal method. <code>trapz
(</code><var>y</var><code>)</code> computes the integral of the <var>y</var> along the first
non-singleton dimension. If the argument <var>x</var> is omitted a
equally spaced vector is assumed. <code>trapz (</code><var>x</var><code>, </code><var>y</var><code>)</code>
evaluates the integral with respect to <var>x</var>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dcumtrapz.html#doc_002dcumtrapz">cumtrapz</a>.
</p></blockquote></div>
<!-- ./general/cumtrapz.m -->
<p><a name="doc_002dcumtrapz"></a>
<div class="defun">
— Function File: <var>z</var> = <b>cumtrapz</b> (<var>y</var>)<var><a name="index-cumtrapz-1777"></a></var><br>
— Function File: <var>z</var> = <b>cumtrapz</b> (<var>x, y</var>)<var><a name="index-cumtrapz-1778"></a></var><br>
— Function File: <var>z</var> = <b>cumtrapz</b> (<var><small class="dots">...</small>, dim</var>)<var><a name="index-cumtrapz-1779"></a></var><br>
<blockquote>
<p>Cumulative numerical integration using trapezoidal method.
<code>cumtrapz (</code><var>y</var><code>)</code> computes the cumulative integral of the
<var>y</var> along the first non-singleton dimension. If the argument
<var>x</var> is omitted a equally spaced vector is assumed. <code>cumtrapz
(</code><var>x</var><code>, </code><var>y</var><code>)</code> evaluates the cumulative integral with respect
to <var>x</var>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dtrapz.html#doc_002dtrapz">trapz</a>, <a href="doc_002dcumsum.html#doc_002dcumsum">cumsum</a>.
</p></blockquote></div>
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