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Next: <a href="QAWC-adaptive-integration-for-Cauchy-principal-values.html#QAWC-adaptive-integration-for-Cauchy-principal-values" accesskey="n" rel="next">QAWC adaptive integration for Cauchy principal values</a>, Previous: <a href="QAGP-adaptive-integration-with-known-singular-points.html#QAGP-adaptive-integration-with-known-singular-points" accesskey="p" rel="previous">QAGP adaptive integration with known singular points</a>, Up: <a href="Numerical-Integration.html#Numerical-Integration" accesskey="u" rel="up">Numerical Integration</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="QAGI-adaptive-integration-on-infinite-intervals-1"></a>
<h3 class="section">17.6 QAGI adaptive integration on infinite intervals</h3>
<a name="index-QAGI-quadrature-algorithm"></a>

<dl>
<dt><a name="index-gsl_005fintegration_005fqagi"></a>Function: <em>int</em> <strong>gsl_integration_qagi</strong> <em>(gsl_function * <var>f</var>, double <var>epsabs</var>, double <var>epsrel</var>, size_t <var>limit</var>, gsl_integration_workspace * <var>workspace</var>, double * <var>result</var>, double * <var>abserr</var>)</em></dt>
<dd>
<p>This function computes the integral of the function <var>f</var> over the
infinite interval <em>(-\infty,+\infty)</em>.  The integral is mapped onto the
semi-open interval <em>(0,1]</em> using the transformation <em>x = (1-t)/t</em>,
</p>
<div class="example">
<pre class="example">\int_{-\infty}^{+\infty} dx f(x) = 
     \int_0^1 dt (f((1-t)/t) + f((-1+t)/t))/t^2.
</pre></div>

<p>It is then integrated using the QAGS algorithm.  The normal 21-point
Gauss-Kronrod rule of QAGS is replaced by a 15-point rule, because the
transformation can generate an integrable singularity at the origin.  In
this case a lower-order rule is more efficient.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fintegration_005fqagiu"></a>Function: <em>int</em> <strong>gsl_integration_qagiu</strong> <em>(gsl_function * <var>f</var>, double <var>a</var>, double <var>epsabs</var>, double <var>epsrel</var>, size_t <var>limit</var>, gsl_integration_workspace * <var>workspace</var>, double * <var>result</var>, double * <var>abserr</var>)</em></dt>
<dd>
<p>This function computes the integral of the function <var>f</var> over the
semi-infinite interval <em>(a,+\infty)</em>.  The integral is mapped onto the
semi-open interval <em>(0,1]</em> using the transformation <em>x = a + (1-t)/t</em>,
</p>
<div class="example">
<pre class="example">\int_{a}^{+\infty} dx f(x) = 
     \int_0^1 dt f(a + (1-t)/t)/t^2
</pre></div>

<p>and then integrated using the QAGS algorithm.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fintegration_005fqagil"></a>Function: <em>int</em> <strong>gsl_integration_qagil</strong> <em>(gsl_function * <var>f</var>, double <var>b</var>, double <var>epsabs</var>, double <var>epsrel</var>, size_t <var>limit</var>, gsl_integration_workspace * <var>workspace</var>, double * <var>result</var>, double * <var>abserr</var>)</em></dt>
<dd><p>This function computes the integral of the function <var>f</var> over the
semi-infinite interval <em>(-\infty,b)</em>.  The integral is mapped onto the
semi-open interval <em>(0,1]</em> using the transformation <em>x = b - (1-t)/t</em>,
</p>
<div class="example">
<pre class="example">\int_{-\infty}^{b} dx f(x) = 
     \int_0^1 dt f(b - (1-t)/t)/t^2
</pre></div>

<p>and then integrated using the QAGS algorithm.
</p></dd></dl>




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