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<a name="Functions-of-One-Variable"></a>
<p>
Next: <a rel="next" accesskey="n" href="Orthogonal-Collocation.html#Orthogonal-Collocation">Orthogonal Collocation</a>,
Up: <a rel="up" accesskey="u" href="Numerical-Integration.html#Numerical-Integration">Numerical Integration</a>
<hr>
</div>
<h3 class="section">23.1 Functions of One Variable</h3>
<p>Octave supports five 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>quadv</code><dd>Numerical integration using an adaptive vectorized Simpson's rule.
<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>quadcc</code><dd>Numerical integration using adaptive Clenshaw-Curtis rules.
<br><dt><code>trapz, cumtrapz</code><dd>Numerical integration of data using the trapezoidal method.
</dl>
<p class="noindent">The best quadrature algorithm to use depends on the integrand. If you have
empirical data, rather than a function, the choice is <code>trapz</code> or
<code>cumtrapz</code>. If you are uncertain about the characteristics of the
integrand, <code>quadcc</code> will be the most robust as it can handle
discontinuities, singularities, oscillatory functions, and infinite intervals.
When the integrand is smooth <code>quadgk</code> may be the fastest of the
algorithms.
<p><table summary=""><tr align="left"><th valign="top" width="5%"></th><th valign="top" width="15%">Function </th><th valign="top" width="80%">Characteristics
<br></th></tr><tr align="left"><td valign="top" width="5%"></td><td valign="top" width="15%">quad </td><td valign="top" width="80%">Low accuracy with nonsmooth integrands
<br></td></tr><tr align="left"><td valign="top" width="5%"></td><td valign="top" width="15%">quadv </td><td valign="top" width="80%">Medium accuracy with smooth integrands
<br></td></tr><tr align="left"><td valign="top" width="5%"></td><td valign="top" width="15%">quadl </td><td valign="top" width="80%">Medium accuracy with smooth integrands. Slower than quadgk.
<br></td></tr><tr align="left"><td valign="top" width="5%"></td><td valign="top" width="15%">quadgk </td><td valign="top" width="80%">Medium accuracy (1e^-6–1e^-9) with smooth integrands.
<br></td></tr><tr align="left"><td valign="top" width="5%"></td><td valign="top" width="15%"></td><td valign="top" width="80%">Handles oscillatory functions and infinite bounds
<br></td></tr><tr align="left"><td valign="top" width="5%"></td><td valign="top" width="15%">quadcc </td><td valign="top" width="80%">Low to High accuracy with nonsmooth/smooth integrands
<br></td></tr><tr align="left"><td valign="top" width="5%"></td><td valign="top" width="15%"></td><td valign="top" width="80%">Handles oscillatory functions, singularities, and infinite bounds
<br></td></tr></table>
<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 <code>quad</code> integrator, but is required by the other integrators. In any
case, it makes it much easier to generate a set of points for plotting because
it is possible to call the function with a vector argument to produce a vector
result.
<p>The second step is to call quad with the limits of integration:
<pre class="example"> [q, 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.).
<p>The function "f" can be the string name of a function, a function handle, or
an inline function. These options make it quite easy to do integration
without having to fully define a function in an m-file. For example:
<pre class="example"> # Verify integral (x^3) = x^4/4
f = inline ("x.^3");
quadgk (f, 0, 1)
⇒ 0.25000
# Verify gamma function = (n-1)! for n = 4
f = @(x) x.^3 .* exp (-x);
quadcc (f, 0, Inf)
⇒ 6.0000
</pre>
<!-- quad src/DLD-FUNCTIONS/quad.cc -->
<p><a name="doc_002dquad"></a>
<div class="defun">
— Loadable Function: <var>q</var> = <b>quad</b> (<var>f, a, b</var>)<var><a name="index-quad-2303"></a></var><br>
— Loadable Function: <var>q</var> = <b>quad</b> (<var>f, a, b, tol</var>)<var><a name="index-quad-2304"></a></var><br>
— Loadable Function: <var>q</var> = <b>quad</b> (<var>f, a, b, tol, sing</var>)<var><a name="index-quad-2305"></a></var><br>
— Loadable Function: [<var>q</var>, <var>ier</var>, <var>nfun</var>, <var>err</var>] = <b>quad</b> (<var><small class="dots">...</small></var>)<var><a name="index-quad-2306"></a></var><br>
<blockquote><p>Numerically evaluate the integral of <var>f</var> from <var>a</var> to <var>b</var> using
Fortran routines from <span class="sc">quadpack</span><!-- /@w -->. <var>f</var> is a function handle,
inline function, or a string containing the name of the function to
evaluate. The function must have the form <code>y = f (x)</code> where <var>y</var> and
<var>x</var> are scalars.
<p><var>a</var> and <var>b</var> are the lower and upper 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. Both tolerances default to <code>sqrt(eps)</code> or
approximately 1.5e^-8.
<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>q</var>. <var>ier</var>
contains an integer error code (0 indicates a successful integration).
<var>nfun</var> indicates the number of function evaluations that were
made, and <var>err</var> contains an estimate of the error in the
solution.
<p>The function <code>quad_options</code> can set other optional
parameters for <code>quad</code>.
<p>Note: because <code>quad</code> is written in Fortran it cannot be called
recursively. This prevents its use in integrating over more than one
variable by routines <code>dblquad</code> and <code>triplequad</code>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dquad_005foptions.html#doc_002dquad_005foptions">quad_options</a>, <a href="doc_002dquadv.html#doc_002dquadv">quadv</a>, <a href="doc_002dquadl.html#doc_002dquadl">quadl</a>, <a href="doc_002dquadgk.html#doc_002dquadgk">quadgk</a>, <a href="doc_002dquadcc.html#doc_002dquadcc">quadcc</a>, <a href="doc_002dtrapz.html#doc_002dtrapz">trapz</a>, <a href="doc_002ddblquad.html#doc_002ddblquad">dblquad</a>, <a href="doc_002dtriplequad.html#doc_002dtriplequad">triplequad</a>.
</p></blockquote></div>
<!-- quad_options src/DLD-FUNCTIONS/quad.cc -->
<p><a name="doc_002dquad_005foptions"></a>
<div class="defun">
— Loadable Function: <b>quad_options</b> ()<var><a name="index-quad_005foptions-2307"></a></var><br>
— Loadable Function: val = <b>quad_options</b> (<var>opt</var>)<var><a name="index-quad_005foptions-2308"></a></var><br>
— Loadable Function: <b>quad_options</b> (<var>opt, val</var>)<var><a name="index-quad_005foptions-2309"></a></var><br>
<blockquote><p>Query or set options for the function <code>quad</code>.
When called with no arguments, the names of all available options and
their current values are displayed.
Given one argument, return the value of the corresponding option.
When called with two arguments, <code>quad_options</code> set the option
<var>opt</var> to value <var>val</var>.
<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>Non-negative 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><!-- /@w -->.
<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>Non-negative 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><!-- /@w -->.
</dl>
</p></blockquote></div>
<!-- quadv scripts/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-2310"></a></var><br>
— Function File: <var>q</var> = <b>quadv</b> (<var>f, a, b, tol</var>)<var><a name="index-quadv-2311"></a></var><br>
— Function File: <var>q</var> = <b>quadv</b> (<var>f, a, b, tol, trace</var>)<var><a name="index-quadv-2312"></a></var><br>
— Function File: <var>q</var> = <b>quadv</b> (<var>f, a, b, tol, trace, p1, p2, <small class="dots">...</small></var>)<var><a name="index-quadv-2313"></a></var><br>
— Function File: [<var>q</var>, <var>nfun</var>] = <b>quadv</b> (<var><small class="dots">...</small></var>)<var><a name="index-quadv-2314"></a></var><br>
<blockquote>
<p>Numerically evaluate the integral of <var>f</var> from <var>a</var> to <var>b</var>
using an adaptive Simpson's rule.
<var>f</var> is a function handle, inline function, or string
containing the name of the function to evaluate.
<code>quadv</code> is a vectorized version of <code>quad</code> and the function
defined by <var>f</var> must accept a scalar or vector as input and return a
scalar, vector, or array as output.
<p><var>a</var> and <var>b</var> are the lower and upper limits of integration. Both
limits must be finite.
<p>The optional argument <var>tol</var> defines the tolerance used to stop
the adaptation procedure. The default value is 1e^-6.
<p>The algorithm used by <code>quadv</code> involves recursively subdividing the
integration interval and applying Simpson's rule on each subinterval.
If <var>trace</var> is true then after computing each of these partial
integrals display: (1) the total number of function evaluations,
(2) the left end of the subinterval, (3) the length of the subinterval,
(4) the approximation of the integral over the subinterval.
<p>Additional arguments <var>p1</var>, etc., are passed directly to the function
<var>f</var>. To use default values for <var>tol</var> and <var>trace</var>, one may pass
empty matrices ([]).
<p>The result of the integration is returned in <var>q</var>. <var>nfun</var> indicates
the number of function evaluations that were made.
<p>Note: <code>quadv</code> is written in Octave's scripting language and can be
used recursively in <code>dblquad</code> and <code>triplequad</code>, unlike the
similar <code>quad</code> function.
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<p class="noindent"><strong>See also:</strong> <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_002dquadcc.html#doc_002dquadcc">quadcc</a>, <a href="doc_002dtrapz.html#doc_002dtrapz">trapz</a>, <a href="doc_002ddblquad.html#doc_002ddblquad">dblquad</a>, <a href="doc_002dtriplequad.html#doc_002dtriplequad">triplequad</a>.
</p></blockquote></div>
<!-- quadl scripts/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-2315"></a></var><br>
— Function File: <var>q</var> = <b>quadl</b> (<var>f, a, b, tol</var>)<var><a name="index-quadl-2316"></a></var><br>
— Function File: <var>q</var> = <b>quadl</b> (<var>f, a, b, tol, trace</var>)<var><a name="index-quadl-2317"></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-2318"></a></var><br>
<blockquote>
<p>Numerically evaluate the integral of <var>f</var> from <var>a</var> to <var>b</var>
using an adaptive Lobatto rule.
<var>f</var> is a function handle, inline function, or string
containing the name of the function to evaluate.
The function <var>f</var> must be vectorized and return a vector of output values
if given a vector of input values.
<p><var>a</var> and <var>b</var> are the lower and upper limits of integration. Both
limits must be finite.
<p>The optional argument <var>tol</var> defines the relative tolerance with which
to perform the integration. The default value is <code>eps</code>.
<p>The algorithm used by <code>quadl</code> involves recursively subdividing the
integration interval.
If <var>trace</var> is defined then for each subinterval display: (1) the left
end of the subinterval, (2) the length of the subinterval, (3) the
approximation of the integral over the subinterval.
<p>Additional arguments <var>p1</var>, etc., are passed directly to the function
<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, <cite>Adaptive Quadrature -
Revisited</cite>, 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>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dquad.html#doc_002dquad">quad</a>, <a href="doc_002dquadv.html#doc_002dquadv">quadv</a>, <a href="doc_002dquadgk.html#doc_002dquadgk">quadgk</a>, <a href="doc_002dquadcc.html#doc_002dquadcc">quadcc</a>, <a href="doc_002dtrapz.html#doc_002dtrapz">trapz</a>, <a href="doc_002ddblquad.html#doc_002ddblquad">dblquad</a>, <a href="doc_002dtriplequad.html#doc_002dtriplequad">triplequad</a>.
</p></blockquote></div>
<!-- quadgk scripts/general/quadgk.m -->
<p><a name="doc_002dquadgk"></a>
<div class="defun">
— Function File: <var>q</var> = <b>quadgk</b> (<var>f, a, b</var>)<var><a name="index-quadgk-2319"></a></var><br>
— Function File: <var>q</var> = <b>quadgk</b> (<var>f, a, b, abstol</var>)<var><a name="index-quadgk-2320"></a></var><br>
— Function File: <var>q</var> = <b>quadgk</b> (<var>f, a, b, abstol, trace</var>)<var><a name="index-quadgk-2321"></a></var><br>
— Function File: <var>q</var> = <b>quadgk</b> (<var>f, a, b, prop, val, <small class="dots">...</small></var>)<var><a name="index-quadgk-2322"></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-2323"></a></var><br>
<blockquote>
<p>Numerically evaluate the integral of <var>f</var> from <var>a</var> to <var>b</var>
using adaptive Gauss-Konrod quadrature.
<var>f</var> is a function handle, inline function, or string
containing the name of the function to evaluate.
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 vectorized and must accept a vector of input values <var>x</var>
and return an output vector representing the function evaluations at the
given values of <var>x</var>.
<p><var>a</var> and <var>b</var> are the lower and upper limits of integration. Either
or both limits may be infinite or contain weak end singularities.
Variable transformation will be used to treat any 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 optional argument <var>tol</var> defines the absolute tolerance used to stop
the integration procedure. The default value is 1e^-10.
<p>The algorithm used by <code>quadgk</code> involves subdividing the
integration interval and evaluating each subinterval.
If <var>trace</var> is true then after computing each of these partial
integrals display: (1) the number of subintervals at this step,
(2) the current estimate of the error <var>err</var>, (3) the current estimate
for the integral <var>q</var>.
<p>Alternatively, properties of <code>quadgk</code> can be passed to the function as
pairs <code>"</code><var>prop</var><code>", </code><var>val</var>. Valid properties are
<dl>
<dt><code>AbsTol</code><dd>Define the absolute error tolerance for the quadrature. The default
absolute tolerance is 1e-10.
<br><dt><code>RelTol</code><dd>Define 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. Subintervals that have an
unacceptable error are subdivided and re-evaluated. If the number of
subintervals exceeds 650 subintervals at any point 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 subintervals that can exist before exiting.
<br><dt><code>WayPoints</code><dd>Discontinuities in the first derivative of the function to integrate can be
flagged with the <code>"WayPoints"</code> property. This forces the ends of
a subinterval 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 <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>The result of the integration is returned in <var>q</var>.
<var>err</var> is an approximate bound 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_002dquad.html#doc_002dquad">quad</a>, <a href="doc_002dquadv.html#doc_002dquadv">quadv</a>, <a href="doc_002dquadl.html#doc_002dquadl">quadl</a>, <a href="doc_002dquadcc.html#doc_002dquadcc">quadcc</a>, <a href="doc_002dtrapz.html#doc_002dtrapz">trapz</a>, <a href="doc_002ddblquad.html#doc_002ddblquad">dblquad</a>, <a href="doc_002dtriplequad.html#doc_002dtriplequad">triplequad</a>.
</p></blockquote></div>
<!-- quadcc src/DLD-FUNCTIONS/quadcc.cc -->
<p><a name="doc_002dquadcc"></a>
<div class="defun">
— Function File: <var>q</var> = <b>quadcc</b> (<var>f, a, b</var>)<var><a name="index-quadcc-2324"></a></var><br>
— Function File: <var>q</var> = <b>quadcc</b> (<var>f, a, b, tol</var>)<var><a name="index-quadcc-2325"></a></var><br>
— Function File: <var>q</var> = <b>quadcc</b> (<var>f, a, b, tol, sing</var>)<var><a name="index-quadcc-2326"></a></var><br>
— Function File: [<var>q</var>, <var>err</var>, <var>nr_points</var>] = <b>quadcc</b> (<var><small class="dots">...</small></var>)<var><a name="index-quadcc-2327"></a></var><br>
<blockquote><p>Numerically evaluate the integral of <var>f</var> from <var>a</var> to <var>b</var>
using the doubly-adaptive Clenshaw-Curtis quadrature described by P. Gonnet
in <cite>Increasing the Reliability of Adaptive Quadrature Using Explicit
Interpolants</cite>.
<var>f</var> is a function handle, inline function, or string
containing the name of the function to evaluate.
The function <var>f</var> must be vectorized and must return a vector of output
values if given a vector of input values. For example,
<pre class="example"> f = @(x) x .* sin (1./x) .* sqrt (abs (1 - x));
</pre>
<p class="noindent">which uses the element-by-element `dot' form for all operators.
<p><var>a</var> and <var>b</var> are the lower and upper limits of integration. Either
or both limits may be infinite. <code>quadcc</code> handles an inifinite limit
by substituting the variable of integration with <code>x=tan(pi/2*u)</code>.
<p>The optional argument <var>tol</var> defines the relative tolerance used to stop
the integration procedure. The default value is 1e^-6.
<p>The optional argument <var>sing</var> contains a list of points where the
integrand has known singularities, or discontinuities
in any of its derivatives, inside the integration interval.
For the example above, which has a discontinuity at x=1, the call to
<code>quadcc</code> would be as follows
<pre class="example"> int = quadcc (f, a, b, 1.0e-6, [ 1 ]);
</pre>
<p>The result of the integration is returned in <var>q</var>.
<var>err</var> is an estimate of the absolute integration error and
<var>nr_points</var> is the number of points at which the integrand was evaluated.
If the adaptive integration did not converge, the value of
<var>err</var> will be larger than the requested tolerance. Therefore, it is
recommended to verify this value for difficult integrands.
<p><code>quadcc</code> is capable of dealing with non-numeric
values of the integrand such as <code>NaN</code> or <code>Inf</code>.
If the integral diverges, and <code>quadcc</code> detects this,
then a warning is issued and <code>Inf</code> or <code>-Inf</code> is returned.
<p>Note: <code>quadcc</code> is a general purpose quadrature algorithm
and, as such, may be less efficient for a smooth or otherwise
well-behaved integrand than other methods such as <code>quadgk</code>.
<p>The algorithm uses Clenshaw-Curtis quadrature rules of increasing
degree in each interval and bisects the interval if either the
function does not appear to be smooth or a rule of maximum
degree has been reached. The error estimate is computed from the
L2-norm of the difference between two successive interpolations
of the integrand over the nodes of the respective quadrature rules.
<p>Reference: P. Gonnet, <cite>Increasing the Reliability of Adaptive
Quadrature Using Explicit Interpolants</cite>, ACM Transactions on
Mathematical Software, Vol. 37, Issue 3, Article No. 3, 2010.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dquad.html#doc_002dquad">quad</a>, <a href="doc_002dquadv.html#doc_002dquadv">quadv</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>, <a href="doc_002ddblquad.html#doc_002ddblquad">dblquad</a>, <a href="doc_002dtriplequad.html#doc_002dtriplequad">triplequad</a>.
</p></blockquote></div>
<p>Sometimes one does not have the function, but only the raw (x, y) points from
which to perform an integration. This can occur when collecting data in an
experiment. The <code>trapz</code> function can integrate these values as shown in
the following example where "data" has been collected on the cosine function
over the range [0, pi/2).
<pre class="example"> x = 0:0.1:pi/2; # Uniformly spaced points
y = cos (x);
trapz (x, y)
⇒ 0.99666
</pre>
<p>The answer is reasonably close to the exact value of 1. Ordinary quadrature
is sensitive to the characteristics of the integrand. Empirical integration
depends not just on the integrand, but also on the particular points chosen to
represent the function. Repeating the example above with the sine function
over the range [0, pi/2) produces far inferior results.
<pre class="example"> x = 0:0.1:pi/2; # Uniformly spaced points
y = sin (x);
trapz (x, y)
⇒ 0.92849
</pre>
<p>However, a slightly different choice of data points can change the result
significantly. The same integration, with the same number of points, but
spaced differently produces a more accurate answer.
<pre class="example"> x = linspace (0, pi/2, 16); # Uniformly spaced, but including endpoint
y = sin (x);
trapz (x, y)
⇒ 0.99909
</pre>
<p>In general there may be no way of knowing the best distribution of points ahead
of time. Or the points may come from an experiment where there is no freedom to
select the best distribution. In any case, one must remain aware of this
issue when using <code>trapz</code>.
<!-- trapz scripts/general/trapz.m -->
<p><a name="doc_002dtrapz"></a>
<div class="defun">
— Function File: <var>q</var> = <b>trapz</b> (<var>y</var>)<var><a name="index-trapz-2328"></a></var><br>
— Function File: <var>q</var> = <b>trapz</b> (<var>x, y</var>)<var><a name="index-trapz-2329"></a></var><br>
— Function File: <var>q</var> = <b>trapz</b> (<var><small class="dots">...</small>, dim</var>)<var><a name="index-trapz-2330"></a></var><br>
<blockquote>
<p>Numerically evaluate the integral of points <var>y</var> using the trapezoidal
method.
<code>trapz (</code><var>y</var><code>)</code><!-- /@w --> computes the integral of <var>y</var> along the first
non-singleton dimension. When the argument <var>x</var> is omitted an
equally spaced <var>x</var> vector with unit spacing (1) is assumed.
<code>trapz (</code><var>x</var><code>, </code><var>y</var><code>)</code> evaluates the integral with respect
to the spacing in <var>x</var> and the values in <var>y</var>. This is useful if
the points in <var>y</var> have been sampled unevenly.
If the optional <var>dim</var> argument is given, operate along this dimension.
<p>If <var>x</var> is not specified then unit spacing will be used. To scale
the integral to the correct value you must multiply by the actual spacing
value (deltaX). As an example, the integral of x^3 over the range
[0, 1] is x^4/4 or 0.25. The following code uses <code>trapz</code> to
calculate the integral in three different ways.
<pre class="example"> x = 0:0.1:1;
y = x.^3;
q = trapz (y)
⇒ q = 2.525 # No scaling
q * 0.1
⇒ q = 0.2525 # Approximation to integral by scaling
trapz (x, y)
⇒ q = 0.2525 # Same result by specifying <var>x</var>
</pre>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dcumtrapz.html#doc_002dcumtrapz">cumtrapz</a>.
</p></blockquote></div>
<!-- cumtrapz scripts/general/cumtrapz.m -->
<p><a name="doc_002dcumtrapz"></a>
<div class="defun">
— Function File: <var>q</var> = <b>cumtrapz</b> (<var>y</var>)<var><a name="index-cumtrapz-2331"></a></var><br>
— Function File: <var>q</var> = <b>cumtrapz</b> (<var>x, y</var>)<var><a name="index-cumtrapz-2332"></a></var><br>
— Function File: <var>q</var> = <b>cumtrapz</b> (<var><small class="dots">...</small>, dim</var>)<var><a name="index-cumtrapz-2333"></a></var><br>
<blockquote>
<p>Cumulative numerical integration of points <var>y</var> using the trapezoidal
method.
<code>cumtrapz (</code><var>y</var><code>)</code><!-- /@w --> computes the cumulative integral of <var>y</var>
along the first non-singleton dimension. Where <code>trapz</code> reports
only the overall integral sum, <code>cumtrapz</code> reports the current partial
sum value at each point of <var>y</var>. When the argument <var>x</var> is omitted
an equally spaced <var>x</var> vector with unit spacing (1) is assumed.
<code>cumtrapz (</code><var>x</var><code>, </code><var>y</var><code>)</code> evaluates the integral with respect to
the spacing in <var>x</var> and the values in <var>y</var>. This is useful if the
points in <var>y</var> have been sampled unevenly. If the optional <var>dim</var>
argument is given, operate along this dimension.
<p>If <var>x</var> is not specified then unit spacing will be used. To scale
the integral to the correct value you must multiply by the actual spacing
value (deltaX).
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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>
</body></html>
