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<a name="Functions-of-Multiple-Variables"></a>
<div class="header">
<p>
Previous: <a href="Orthogonal-Collocation.html#Orthogonal-Collocation" accesskey="p" rel="prev">Orthogonal Collocation</a>, Up: <a href="Numerical-Integration.html#Numerical-Integration" accesskey="u" rel="up">Numerical Integration</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="Functions-of-Multiple-Variables-1"></a>
<h3 class="section">23.3 Functions of Multiple Variables</h3>
<p>Octave includes several functions for computing the integral of functions of
multiple variables. This procedure can generally be performed by creating a
function that integrates <em>f</em> with respect to <em>x</em>, and then integrates
that function with respect to <em>y</em>. This procedure can be performed
manually using the following example which integrates the function:
</p>
<div class="example">
<pre class="example">f(x, y) = sin(pi*x*y) * sqrt(x*y)
</pre></div>
<p>for <em>x</em> and <em>y</em> between 0 and 1.
</p>
<p>Using <code>quadgk</code> in the example below, a double integration can be
performed. (Note that any of the 1-D quadrature functions can be used in this
fashion except for <code>quad</code> since it is written in Fortran and cannot be
called recursively.)
</p>
<div class="example">
<pre class="example">function q = g(y)
q = ones (size (y));
for i = 1:length (y)
f = @(x) sin (pi*x.*y(i)) .* sqrt (x.*y(i));
q(i) = quadgk (f, 0, 1);
endfor
endfunction
I = quadgk ("g", 0, 1)
⇒ 0.30022
</pre></div>
<p>The algorithm above is implemented in the function <code>dblquad</code> for integrals
over two variables. The 3-D equivalent of this process is implemented in
<code>triplequad</code> for integrals over three variables. As an example, the
result above can be replicated with a call to <code>dblquad</code> as shown below.
</p>
<div class="example">
<pre class="example">I = dblquad (@(x, y) sin (pi*x.*y) .* sqrt (x.*y), 0, 1, 0, 1)
⇒ 0.30022
</pre></div>
<a name="XREFdblquad"></a><dl>
<dt><a name="index-dblquad"></a><em></em> <strong>dblquad</strong> <em>(<var>f</var>, <var>xa</var>, <var>xb</var>, <var>ya</var>, <var>yb</var>)</em></dt>
<dt><a name="index-dblquad-1"></a><em></em> <strong>dblquad</strong> <em>(<var>f</var>, <var>xa</var>, <var>xb</var>, <var>ya</var>, <var>yb</var>, <var>tol</var>)</em></dt>
<dt><a name="index-dblquad-2"></a><em></em> <strong>dblquad</strong> <em>(<var>f</var>, <var>xa</var>, <var>xb</var>, <var>ya</var>, <var>yb</var>, <var>tol</var>, <var>quadf</var>)</em></dt>
<dt><a name="index-dblquad-3"></a><em></em> <strong>dblquad</strong> <em>(<var>f</var>, <var>xa</var>, <var>xb</var>, <var>ya</var>, <var>yb</var>, <var>tol</var>, <var>quadf</var>, …)</em></dt>
<dd><p>Numerically evaluate the double integral of <var>f</var>.
</p>
<p><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 have the form
<em>z = f(x,y)</em> where <var>x</var> is a vector and <var>y</var> is a scalar. It
should return a vector of the same length and orientation as <var>x</var>.
</p>
<p><var>xa</var>, <var>ya</var> and <var>xb</var>, <var>yb</var> are the lower and upper limits of
integration for x and y respectively. The underlying integrator determines
whether infinite bounds are accepted.
</p>
<p>The optional argument <var>tol</var> defines the absolute tolerance used to
integrate each sub-integral. The default value is 1e-6.
</p>
<p>The optional argument <var>quadf</var> specifies which underlying integrator
function to use. Any choice but <code>quad</code> is available and the default
is <code>quadcc</code>.
</p>
<p>Additional arguments, are passed directly to <var>f</var>. To use the default
value for <var>tol</var> or <var>quadf</var> one may pass <code>':'</code> or an empty
matrix ([]).
</p>
<p><strong>See also:</strong> <a href="#XREFintegral2">integral2</a>, <a href="#XREFintegral3">integral3</a>, <a href="#XREFtriplequad">triplequad</a>, <a href="Functions-of-One-Variable.html#XREFquad">quad</a>, <a href="Functions-of-One-Variable.html#XREFquadv">quadv</a>, <a href="Functions-of-One-Variable.html#XREFquadl">quadl</a>, <a href="Functions-of-One-Variable.html#XREFquadgk">quadgk</a>, <a href="Functions-of-One-Variable.html#XREFquadcc">quadcc</a>, <a href="Functions-of-One-Variable.html#XREFtrapz">trapz</a>.
</p></dd></dl>
<a name="XREFtriplequad"></a><dl>
<dt><a name="index-triplequad"></a><em></em> <strong>triplequad</strong> <em>(<var>f</var>, <var>xa</var>, <var>xb</var>, <var>ya</var>, <var>yb</var>, <var>za</var>, <var>zb</var>)</em></dt>
<dt><a name="index-triplequad-1"></a><em></em> <strong>triplequad</strong> <em>(<var>f</var>, <var>xa</var>, <var>xb</var>, <var>ya</var>, <var>yb</var>, <var>za</var>, <var>zb</var>, <var>tol</var>)</em></dt>
<dt><a name="index-triplequad-2"></a><em></em> <strong>triplequad</strong> <em>(<var>f</var>, <var>xa</var>, <var>xb</var>, <var>ya</var>, <var>yb</var>, <var>za</var>, <var>zb</var>, <var>tol</var>, <var>quadf</var>)</em></dt>
<dt><a name="index-triplequad-3"></a><em></em> <strong>triplequad</strong> <em>(<var>f</var>, <var>xa</var>, <var>xb</var>, <var>ya</var>, <var>yb</var>, <var>za</var>, <var>zb</var>, <var>tol</var>, <var>quadf</var>, …)</em></dt>
<dd><p>Numerically evaluate the triple integral of <var>f</var>.
</p>
<p><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 have the form
<em>w = f(x,y,z)</em> where either <var>x</var> or <var>y</var> is a vector and the
remaining inputs are scalars. It should return a vector of the same length
and orientation as <var>x</var> or <var>y</var>.
</p>
<p><var>xa</var>, <var>ya</var>, <var>za</var> and <var>xb</var>, <var>yb</var>, <var>zb</var> are the lower
and upper limits of integration for x, y, and z respectively. The
underlying integrator determines whether infinite bounds are accepted.
</p>
<p>The optional argument <var>tol</var> defines the absolute tolerance used to
integrate each sub-integral. The default value is 1e-6.
</p>
<p>The optional argument <var>quadf</var> specifies which underlying integrator
function to use. Any choice but <code>quad</code> is available and the default
is <code>quadcc</code>.
</p>
<p>Additional arguments, are passed directly to <var>f</var>. To use the default
value for <var>tol</var> or <var>quadf</var> one may pass <code>':'</code> or an empty
matrix ([]).
</p>
<p><strong>See also:</strong> <a href="#XREFintegral3">integral3</a>, <a href="#XREFintegral2">integral2</a>, <a href="#XREFdblquad">dblquad</a>, <a href="Functions-of-One-Variable.html#XREFquad">quad</a>, <a href="Functions-of-One-Variable.html#XREFquadv">quadv</a>, <a href="Functions-of-One-Variable.html#XREFquadl">quadl</a>, <a href="Functions-of-One-Variable.html#XREFquadgk">quadgk</a>, <a href="Functions-of-One-Variable.html#XREFquadcc">quadcc</a>, <a href="Functions-of-One-Variable.html#XREFtrapz">trapz</a>.
</p></dd></dl>
<p>The recursive algorithm for quadrature presented above is referred to as
<code>"iterated"</code>. A separate 2-D integration method is implemented in the
function <code>quad2d</code>. This function performs a <code>"tiled"</code> integration
by subdividing the integration domain into rectangular regions and performing
separate integrations over those domains. The domains are further subdivided
in areas requiring refinement to reach the desired numerical accuracy. For
certain functions this method can be faster than the 2-D iteration used in the
other functions above.
</p>
<a name="XREFquad2d"></a><dl>
<dt><a name="index-quad2d"></a><em><var>q</var> =</em> <strong>quad2d</strong> <em>(<var>f</var>, <var>xa</var>, <var>xb</var>, <var>ya</var>, <var>yb</var>)</em></dt>
<dt><a name="index-quad2d-1"></a><em><var>q</var> =</em> <strong>quad2d</strong> <em>(<var>f</var>, <var>xa</var>, <var>xb</var>, <var>ya</var>, <var>yb</var>, <var>prop</var>, <var>val</var>, …)</em></dt>
<dt><a name="index-quad2d-2"></a><em>[<var>q</var>, <var>err</var>, <var>iter</var>] =</em> <strong>quad2d</strong> <em>(…)</em></dt>
<dd>
<p>Numerically evaluate the two-dimensional integral of <var>f</var> using adaptive
quadrature over the two-dimensional domain defined by <var>xa</var>, <var>xb</var>,
<var>ya</var>, <var>yb</var> using tiled integration. Additionally, <var>ya</var> and
<var>yb</var> may be scalar functions of <var>x</var>, allowing for the integration
over non-rectangular domains.
</p>
<p><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 of the form
<em>z = f(x,y)</em> where <var>x</var> is a vector and <var>y</var> is a scalar. It
should return a vector of the same length and orientation as <var>x</var>.
</p>
<p>Additional optional parameters can be specified using
<code>"<var>property</var>", <var>value</var></code> pairs. Valid properties are:
</p>
<dl compact="compact">
<dt><code>AbsTol</code></dt>
<dd><p>Define the absolute error tolerance for the quadrature. The default
value is 1e-10 (1e-5 for single).
</p>
</dd>
<dt><code>RelTol</code></dt>
<dd><p>Define the relative error tolerance for the quadrature. The default
value is 1e-6 (1e-4 for single).
</p>
</dd>
<dt><code>MaxFunEvals</code></dt>
<dd><p>The maximum number of function calls to the vectorized function <var>f</var>.
The default value is 5000.
</p>
</dd>
<dt><code>Singular</code></dt>
<dd><p>Enable/disable transforms to weaken singularities on the edge of the
integration domain. The default value is <var>true</var>.
</p>
</dd>
<dt><code>Vectorized</code></dt>
<dd><p>Option to disable vectorized integration, forcing Octave to use only scalar
inputs when calling the integrand. The default value is <var>false</var>.
</p>
</dd>
<dt><code>FailurePlot</code></dt>
<dd><p>If <code>quad2d</code> fails to converge to the desired error tolerance before
MaxFunEvals is reached, a plot of the areas that still need refinement
is created. The default value is <var>false</var>.
</p></dd>
</dl>
<p>Adaptive quadrature is used to minimize the estimate of error until the
following is satisfied:
</p>
<div class="example">
<pre class="example"> <var>error</var> <= max (<var>AbsTol</var>, <var>RelTol</var>*|<var>q</var>|)
</pre></div>
<p>The optional output <var>err</var> is an approximate bound on the error in the
integral <code>abs (<var>q</var> - <var>I</var>)</code>, where <var>I</var> is the exact value
of the integral. The optional output <var>iter</var> is the number of vectorized
function calls to the function <var>f</var> that were used.
</p>
<p>Example 1 : integrate a rectangular region in x-y plane
</p>
<div class="example">
<pre class="example"><var>f</var> = @(<var>x</var>,<var>y</var>) 2*ones (size (<var>x</var>));
<var>q</var> = quad2d (<var>f</var>, 0, 1, 0, 1)
⇒ <var>q</var> = 2
</pre></div>
<p>The result is a volume, which for this constant-value integrand, is just
<code><var>Length</var> * <var>Width</var> * <var>Height</var></code>.
</p>
<p>Example 2 : integrate a triangular region in x-y plane
</p>
<div class="example">
<pre class="example"><var>f</var> = @(<var>x</var>,<var>y</var>) 2*ones (size (<var>x</var>));
<var>ymax</var> = @(<var>x</var>) 1 - <var>x</var>;
<var>q</var> = quad2d (<var>f</var>, 0, 1, 0, <var>ymax</var>)
⇒ <var>q</var> = 1
</pre></div>
<p>The result is a volume, which for this constant-value integrand, is the
Triangle Area x Height or
<code>1/2 * <var>Base</var> * <var>Width</var> * <var>Height</var></code>.
</p>
<p>Programming Notes: If there are singularities within the integration region
it is best to split the integral and place the singularities on the
boundary.
</p>
<p>Known <small>MATLAB</small> incompatibility: If tolerances are left unspecified, and
any integration limits are of type <code>single</code>, then Octave’s integral
functions automatically reduce the default absolute and relative error
tolerances as specified above. If tighter tolerances are desired they
must be specified. <small>MATLAB</small> leaves the tighter tolerances appropriate
for <code>double</code> inputs in place regardless of the class of the
integration limits.
</p>
<p>Reference: L.F. Shampine,
<cite><small>MATLAB</small> program for quadrature in 2D</cite>, Applied Mathematics and
Computation, pp. 266–274, Vol 1, 2008.
</p>
<p><strong>See also:</strong> <a href="#XREFintegral2">integral2</a>, <a href="#XREFdblquad">dblquad</a>, <a href="Functions-of-One-Variable.html#XREFintegral">integral</a>, <a href="Functions-of-One-Variable.html#XREFquad">quad</a>, <a href="Functions-of-One-Variable.html#XREFquadgk">quadgk</a>, <a href="Functions-of-One-Variable.html#XREFquadv">quadv</a>, <a href="Functions-of-One-Variable.html#XREFquadl">quadl</a>, <a href="Functions-of-One-Variable.html#XREFquadcc">quadcc</a>, <a href="Functions-of-One-Variable.html#XREFtrapz">trapz</a>, <a href="#XREFintegral3">integral3</a>, <a href="#XREFtriplequad">triplequad</a>.
</p></dd></dl>
<p>Finally, the functions <code>integral2</code> and <code>integral3</code> are provided
as general 2-D and 3-D integration functions. They will auto-select between
iterated and tiled integration methods and, unlike <code>dblquad</code> and
<code>triplequad</code>, will work with non-rectangular integration domains.
</p>
<a name="XREFintegral2"></a><dl>
<dt><a name="index-integral2"></a><em><var>q</var> =</em> <strong>integral2</strong> <em>(<var>f</var>, <var>xa</var>, <var>xb</var>, <var>ya</var>, <var>yb</var>)</em></dt>
<dt><a name="index-integral2-1"></a><em><var>q</var> =</em> <strong>integral2</strong> <em>(<var>f</var>, <var>xa</var>, <var>xb</var>, <var>ya</var>, <var>yb</var>, <var>prop</var>, <var>val</var>, …)</em></dt>
<dt><a name="index-integral2-2"></a><em>[<var>q</var>, <var>err</var>] =</em> <strong>integral2</strong> <em>(…)</em></dt>
<dd>
<p>Numerically evaluate the two-dimensional integral of <var>f</var> using adaptive
quadrature over the two-dimensional domain defined by <var>xa</var>, <var>xb</var>,
<var>ya</var>, <var>yb</var> (scalars may be finite or infinite). Additionally,
<var>ya</var> and <var>yb</var> may be scalar functions of <var>x</var>, allowing for
integration over non-rectangular domains.
</p>
<p><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 of the form
<em>z = f(x,y)</em> where <var>x</var> is a vector and <var>y</var> is a scalar. It
should return a vector of the same length and orientation as <var>x</var>.
</p>
<p>Additional optional parameters can be specified using
<code>"<var>property</var>", <var>value</var></code> pairs. Valid properties are:
</p>
<dl compact="compact">
<dt><code>AbsTol</code></dt>
<dd><p>Define the absolute error tolerance for the quadrature. The default
value is 1e-10 (1e-5 for single).
</p>
</dd>
<dt><code>RelTol</code></dt>
<dd><p>Define the relative error tolerance for the quadrature. The default
value is 1e-6 (1e-4 for single).
</p>
</dd>
<dt><code>Method</code></dt>
<dd><p>Specify the two-dimensional integration method to be used, with valid
options being <code>"auto"</code> (default), <code>"tiled"</code>, or
<code>"iterated"</code>. When using <code>"auto"</code>, Octave will choose the
<code>"tiled"</code> method unless any of the integration limits are infinite.
</p>
</dd>
<dt><code>Vectorized</code></dt>
<dd><p>Enable or disable vectorized integration. A value of <code>false</code> forces
Octave to use only scalar inputs when calling the integrand, which enables
integrands <em>f(x,y)</em> that have not been vectorized and only accept
<var>x</var> and <var>y</var> as scalars to be used. The default value is
<code>true</code>.
</p></dd>
</dl>
<p>Adaptive quadrature is used to minimize the estimate of error until the
following is satisfied:
</p>
<div class="example">
<pre class="example"> <var>error</var> <= max (<var>AbsTol</var>, <var>RelTol</var>*|<var>q</var>|)
</pre></div>
<p><var>err</var> is an approximate bound on the error in the integral
<code>abs (<var>q</var> - <var>I</var>)</code>, where <var>I</var> is the exact value of the
integral.
</p>
<p>Example 1 : integrate a rectangular region in x-y plane
</p>
<div class="example">
<pre class="example"><var>f</var> = @(<var>x</var>,<var>y</var>) 2*ones (size (<var>x</var>));
<var>q</var> = integral2 (<var>f</var>, 0, 1, 0, 1)
⇒ <var>q</var> = 2
</pre></div>
<p>The result is a volume, which for this constant-value integrand, is just
<code><var>Length</var> * <var>Width</var> * <var>Height</var></code>.
</p>
<p>Example 2 : integrate a triangular region in x-y plane
</p>
<div class="example">
<pre class="example"><var>f</var> = @(<var>x</var>,<var>y</var>) 2*ones (size (<var>x</var>));
<var>ymax</var> = @(<var>x</var>) 1 - <var>x</var>;
<var>q</var> = integral2 (<var>f</var>, 0, 1, 0, <var>ymax</var>)
⇒ <var>q</var> = 1
</pre></div>
<p>The result is a volume, which for this constant-value integrand, is the
Triangle Area x Height or
<code>1/2 * <var>Base</var> * <var>Width</var> * <var>Height</var></code>.
</p>
<p>Programming Notes: If there are singularities within the integration region
it is best to split the integral and place the singularities on the
boundary.
</p>
<p>Known <small>MATLAB</small> incompatibility: If tolerances are left unspecified, and
any integration limits are of type <code>single</code>, then Octave’s integral
functions automatically reduce the default absolute and relative error
tolerances as specified above. If tighter tolerances are desired they
must be specified. <small>MATLAB</small> leaves the tighter tolerances appropriate
for <code>double</code> inputs in place regardless of the class of the
integration limits.
</p>
<p>Reference: L.F. Shampine,
<cite><small>MATLAB</small> program for quadrature in 2D</cite>, Applied Mathematics and
Computation, pp. 266–274, Vol 1, 2008.
</p>
<p><strong>See also:</strong> <a href="#XREFquad2d">quad2d</a>, <a href="#XREFdblquad">dblquad</a>, <a href="Functions-of-One-Variable.html#XREFintegral">integral</a>, <a href="Functions-of-One-Variable.html#XREFquad">quad</a>, <a href="Functions-of-One-Variable.html#XREFquadgk">quadgk</a>, <a href="Functions-of-One-Variable.html#XREFquadv">quadv</a>, <a href="Functions-of-One-Variable.html#XREFquadl">quadl</a>, <a href="Functions-of-One-Variable.html#XREFquadcc">quadcc</a>, <a href="Functions-of-One-Variable.html#XREFtrapz">trapz</a>, <a href="#XREFintegral3">integral3</a>, <a href="#XREFtriplequad">triplequad</a>.
</p></dd></dl>
<a name="XREFintegral3"></a><dl>
<dt><a name="index-integral3"></a><em><var>q</var> =</em> <strong>integral3</strong> <em>(<var>f</var>, <var>xa</var>, <var>xb</var>, <var>ya</var>, <var>yb</var>, <var>za</var>, <var>zb</var>)</em></dt>
<dt><a name="index-integral3-1"></a><em><var>q</var> =</em> <strong>integral3</strong> <em>(<var>f</var>, <var>xa</var>, <var>xb</var>, <var>ya</var>, <var>yb</var>, <var>za</var>, <var>zb</var>, <var>prop</var>, <var>val</var>, …)</em></dt>
<dd>
<p>Numerically evaluate the three-dimensional integral of <var>f</var> using
adaptive quadrature over the three-dimensional domain defined by
<var>xa</var>, <var>xb</var>, <var>ya</var>, <var>yb</var>, <var>za</var>, <var>zb</var> (scalars may
be finite or infinite). Additionally, <var>ya</var> and <var>yb</var> may be
scalar functions of <var>x</var> and <var>za</var>, and <var>zb</var> maybe be scalar
functions of <var>x</var> and <var>y</var>, allowing for integration over
non-rectangular domains.
</p>
<p><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 of the form
<em>z = f(x,y)</em> where <var>x</var> is a vector and <var>y</var> is a scalar. It
should return a vector of the same length and orientation as <var>x</var>.
</p>
<p>Additional optional parameters can be specified using
<code>"<var>property</var>", <var>value</var></code> pairs. Valid properties are:
</p>
<dl compact="compact">
<dt><code>AbsTol</code></dt>
<dd><p>Define the absolute error tolerance for the quadrature. The default
value is 1e-10 (1e-5 for single).
</p>
</dd>
<dt><code>RelTol</code></dt>
<dd><p>Define the relative error tolerance for the quadrature. The default
value is 1e-6 (1e-4 for single).
</p>
</dd>
<dt><code>Method</code></dt>
<dd><p>Specify the two-dimensional integration method to be used, with valid
options being <code>"auto"</code> (default), <code>"tiled"</code>, or
<code>"iterated"</code>. When using <code>"auto"</code>, Octave will choose the
<code>"tiled"</code> method unless any of the integration limits are infinite.
</p>
</dd>
<dt><code>Vectorized</code></dt>
<dd><p>Enable or disable vectorized integration. A value of <code>false</code> forces
Octave to use only scalar inputs when calling the integrand, which enables
integrands <em>f(x,y)</em> that have not been vectorized and only accept
<var>x</var> and <var>y</var> as scalars to be used. The default value is
<code>true</code>.
</p></dd>
</dl>
<p>Adaptive quadrature is used to minimize the estimate of error until the
following is satisfied:
</p>
<div class="example">
<pre class="example"> <var>error</var> <= max (<var>AbsTol</var>, <var>RelTol</var>*|<var>q</var>|)
</pre></div>
<p><var>err</var> is an approximate bound on the error in the integral
<code>abs (<var>q</var> - <var>I</var>)</code>, where <var>I</var> is the exact value of the
integral.
</p>
<p>Example 1 : integrate over a rectangular volume
</p>
<div class="example">
<pre class="example"><var>f</var> = @(<var>x</var>,<var>y</var>,<var>z</var>) ones (size (<var>x</var>));
<var>q</var> = integral3 (<var>f</var>, 0, 1, 0, 1, 0, 1)
⇒ <var>q</var> = 1.00000
</pre></div>
<p>For this constant-value integrand, the result is a volume which is just
<code><var>Length</var> * <var>Width</var> * <var>Height</var></code>.
</p>
<p>Example 2 : integrate over a spherical volume
</p>
<div class="example">
<pre class="example"><var>f</var> = @(<var>x</var>,<var>y</var>) ones (size (<var>x</var>));
<var>ymax</var> = @(<var>x</var>) sqrt (1 - <var>x</var>.^2);
<var>zmax</var> = @(<var>x</var>,<var>y</var>) sqrt (1 - <var>x</var>.^2 - <var>y</var>.^2);
<var>q</var> = integral3 (<var>f</var>, 0, 1, 0, <var>ymax</var>, 0, <var>zmax</var>)
⇒ <var>q</var> = 0.52360
</pre></div>
<p>For this constant-value integrand, the result is a volume which is 1/8th
of a unit sphere or <code>1/8 * 4/3 * pi</code>.
</p>
<p>Programming Notes: If there are singularities within the integration region
it is best to split the integral and place the singularities on the
boundary.
</p>
<p>Known <small>MATLAB</small> incompatibility: If tolerances are left unspecified, and
any integration limits are of type <code>single</code>, then Octave’s integral
functions automatically reduce the default absolute and relative error
tolerances as specified above. If tighter tolerances are desired they
must be specified. <small>MATLAB</small> leaves the tighter tolerances appropriate
for <code>double</code> inputs in place regardless of the class of the
integration limits.
</p>
<p>Reference: L.F. Shampine,
<cite><small>MATLAB</small> program for quadrature in 2D</cite>, Applied Mathematics and
Computation, pp. 266–274, Vol 1, 2008.
</p>
<p><strong>See also:</strong> <a href="#XREFtriplequad">triplequad</a>, <a href="Functions-of-One-Variable.html#XREFintegral">integral</a>, <a href="Functions-of-One-Variable.html#XREFquad">quad</a>, <a href="Functions-of-One-Variable.html#XREFquadgk">quadgk</a>, <a href="Functions-of-One-Variable.html#XREFquadv">quadv</a>, <a href="Functions-of-One-Variable.html#XREFquadl">quadl</a>, <a href="Functions-of-One-Variable.html#XREFquadcc">quadcc</a>, <a href="Functions-of-One-Variable.html#XREFtrapz">trapz</a>, <a href="#XREFintegral2">integral2</a>, <a href="#XREFquad2d">quad2d</a>, <a href="#XREFdblquad">dblquad</a>.
</p></dd></dl>
<p>The above integrations can be fairly slow, and that problem increases
exponentially with the dimensionality of the integral. Another possible
solution for 2-D integration is to use Orthogonal Collocation as described in
the previous section (see <a href="Orthogonal-Collocation.html#Orthogonal-Collocation">Orthogonal Collocation</a>). The integral of a
function <em>f(x,y)</em> for <em>x</em> and <em>y</em> between 0 and 1 can be
approximated using <em>n</em> points by
the sum over <code>i=1:n</code> and <code>j=1:n</code> of <code>q(i)*q(j)*f(r(i),r(j))</code>,
where <em>q</em> and <em>r</em> is as returned by <code>colloc (n)</code>. The
generalization to more than two variables is straight forward. The
following code computes the studied integral using <em>n=8</em> points.
</p>
<div class="example">
<pre class="example">f = @(x,y) sin (pi*x*y') .* sqrt (x*y');
n = 8;
[t, ~, ~, q] = colloc (n);
I = q'*f(t,t)*q;
⇒ 0.30022
</pre></div>
<p>It should be noted that the number of points determines the quality
of the approximation. If the integration needs to be performed between
<em>a</em> and <em>b</em>, instead of 0 and 1, then a change of variables is needed.
</p>
<hr>
<div class="header">
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