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<a name="QAWF-adaptive-integration-for-Fourier-integrals"></a>
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<p>
Next: <a href="CQUAD-doubly_002dadaptive-integration.html#CQUAD-doubly_002dadaptive-integration" accesskey="n" rel="next">CQUAD doubly-adaptive integration</a>, Previous: <a href="QAWO-adaptive-integration-for-oscillatory-functions.html#QAWO-adaptive-integration-for-oscillatory-functions" accesskey="p" rel="previous">QAWO adaptive integration for oscillatory functions</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>
</div>
<hr>
<a name="QAWF-adaptive-integration-for-Fourier-integrals-1"></a>
<h3 class="section">17.10 QAWF adaptive integration for Fourier integrals</h3>
<a name="index-QAWF-quadrature-algorithm"></a>
<a name="index-Fourier-integrals_002c-numerical"></a>

<dl>
<dt><a name="index-gsl_005fintegration_005fqawf"></a>Function: <em>int</em> <strong>gsl_integration_qawf</strong> <em>(gsl_function * <var>f</var>, const double <var>a</var>, const double <var>epsabs</var>, const size_t <var>limit</var>, gsl_integration_workspace * <var>workspace</var>, gsl_integration_workspace * <var>cycle_workspace</var>, gsl_integration_qawo_table * <var>wf</var>, double * <var>result</var>, double * <var>abserr</var>)</em></dt>
<dd>
<p>This function attempts to compute a Fourier integral of the function
<var>f</var> over the semi-infinite interval <em>[a,+\infty)</em>.
</p>
<div class="example">
<pre class="example">I = \int_a^{+\infty} dx f(x) sin(omega x)
I = \int_a^{+\infty} dx f(x) cos(omega x)
</pre></div>

<p>The parameter <em>\omega</em> and choice of <em>\sin</em> or <em>\cos</em> is
taken from the table <var>wf</var> (the length <var>L</var> can take any value,
since it is overridden by this function to a value appropriate for the
Fourier integration).  The integral is computed using the QAWO algorithm
over each of the subintervals,
</p>
<div class="example">
<pre class="example">C_1 = [a, a + c]
C_2 = [a + c, a + 2 c]
... = ...
C_k = [a + (k-1) c, a + k c]
</pre></div>

<p>where 
<em>c = (2 floor(|\omega|) + 1) \pi/|\omega|</em>.  The width <em>c</em> is
chosen to cover an odd number of periods so that the contributions from
the intervals alternate in sign and are monotonically decreasing when
<var>f</var> is positive and monotonically decreasing.  The sum of this
sequence of contributions is accelerated using the epsilon-algorithm.
</p>
<p>This function works to an overall absolute tolerance of
<var>abserr</var>.  The following strategy is used: on each interval
<em>C_k</em> the algorithm tries to achieve the tolerance
</p>
<div class="example">
<pre class="example">TOL_k = u_k abserr
</pre></div>

<p>where 
<em>u_k = (1 - p)p^{k-1}</em> and <em>p = 9/10</em>.  
The sum of the geometric series of contributions from each interval
gives an overall tolerance of <var>abserr</var>.
</p>
<p>If the integration of a subinterval leads to difficulties then the
accuracy requirement for subsequent intervals is relaxed,
</p>
<div class="example">
<pre class="example">TOL_k = u_k max(abserr, max_{i&lt;k}{E_i})
</pre></div>

<p>where <em>E_k</em> is the estimated error on the interval <em>C_k</em>.
</p>
<p>The subintervals and their results are stored in the memory provided by
<var>workspace</var>.  The maximum number of subintervals is given by
<var>limit</var>, which may not exceed the allocated size of the workspace.
The integration over each subinterval uses the memory provided by
<var>cycle_workspace</var> as workspace for the QAWO algorithm.
</p>
</dd></dl>

<hr>
<div class="header">
<p>
Next: <a href="CQUAD-doubly_002dadaptive-integration.html#CQUAD-doubly_002dadaptive-integration" accesskey="n" rel="next">CQUAD doubly-adaptive integration</a>, Previous: <a href="QAWO-adaptive-integration-for-oscillatory-functions.html#QAWO-adaptive-integration-for-oscillatory-functions" accesskey="p" rel="previous">QAWO adaptive integration for oscillatory functions</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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