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<title>Numerical Integration Introduction - GNU Scientific Library -- Reference Manual</title>
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<a name="Numerical-Integration-Introduction"></a>
Next: <a rel="next" accesskey="n" href="QNG-non_002dadaptive-Gauss_002dKronrod-integration.html">QNG non-adaptive Gauss-Kronrod integration</a>,
Up: <a rel="up" accesskey="u" href="Numerical-Integration.html">Numerical Integration</a>
<hr>
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<h3 class="section">16.1 Introduction</h3>
<p>Each algorithm computes an approximation to a definite integral of the
form,
<pre class="example"> I = \int_a^b f(x) w(x) dx
</pre>
<p class="noindent">where w(x) is a weight function (for general integrands w(x)=1).
The user provides absolute and relative error bounds
<!-- {$(\hbox{\it epsabs}, \hbox{\it epsrel}\,)$} -->
(epsabs, epsrel) which specify the following accuracy requirement,
<pre class="example"> |RESULT - I| <= max(epsabs, epsrel |I|)
</pre>
<p class="noindent">where
<!-- {$\hbox{\it RESULT}$} -->
RESULT is the numerical approximation obtained by the
algorithm. The algorithms attempt to estimate the absolute error
<!-- {$\hbox{\it ABSERR} = |\hbox{\it RESULT} - I|$} -->
ABSERR = |RESULT - I| in such a way that the following inequality
holds,
<pre class="example"> |RESULT - I| <= ABSERR <= max(epsabs, epsrel |I|)
</pre>
<p class="noindent">The routines will fail to converge if the error bounds are too
stringent, but always return the best approximation obtained up to that
stage.
<p>The algorithms in <span class="sc">quadpack</span> use a naming convention based on the
following letters,
<pre class="display"> <code>Q</code> - quadrature routine
<code>N</code> - non-adaptive integrator
<code>A</code> - adaptive integrator
<code>G</code> - general integrand (user-defined)
<code>W</code> - weight function with integrand
<code>S</code> - singularities can be more readily integrated
<code>P</code> - points of special difficulty can be supplied
<code>I</code> - infinite range of integration
<code>O</code> - oscillatory weight function, cos or sin
<code>F</code> - Fourier integral
<code>C</code> - Cauchy principal value
</pre>
<p class="noindent">The algorithms are built on pairs of quadrature rules, a higher order
rule and a lower order rule. The higher order rule is used to compute
the best approximation to an integral over a small range. The
difference between the results of the higher order rule and the lower
order rule gives an estimate of the error in the approximation.
<ul class="menu">
<li><a accesskey="1" href="Integrands-without-weight-functions.html">Integrands without weight functions</a>
<li><a accesskey="2" href="Integrands-with-weight-functions.html">Integrands with weight functions</a>
<li><a accesskey="3" href="Integrands-with-singular-weight-functions.html">Integrands with singular weight functions</a>
</ul>
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