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<a name="Numerical-Integration-Introduction"></a>
<div class="header">
<p>
Next: <a href="QNG-non_002dadaptive-Gauss_002dKronrod-integration.html#QNG-non_002dadaptive-Gauss_002dKronrod-integration" accesskey="n" rel="next">QNG non-adaptive Gauss-Kronrod integration</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="Introduction-2"></a>
<h3 class="section">17.1 Introduction</h3>

<p>Each algorithm computes an approximation to a definite integral of the
form,
</p>
<div class="example">
<pre class="example">I = \int_a^b f(x) w(x) dx
</pre></div>

<p>where <em>w(x)</em> is a weight function (for general integrands <em>w(x)=1</em>).
The user provides absolute and relative error bounds 
<em>(epsabs, epsrel)</em> which specify the following accuracy requirement,
</p>
<div class="example">
<pre class="example">|RESULT - I|  &lt;= max(epsabs, epsrel |I|)
</pre></div>

<p>where 
<em>RESULT</em> is the numerical approximation obtained by the
algorithm.  The algorithms attempt to estimate the absolute error
<em>ABSERR = |RESULT - I|</em> in such a way that the following inequality
holds,
</p>
<div class="example">
<pre class="example">|RESULT - I| &lt;= ABSERR &lt;= max(epsabs, epsrel |I|)
</pre></div>

<p>In short, the routines return the first approximation 
which has an absolute error smaller than <em>epsabs</em> or a relative error smaller than <em>epsrel</em>.   
</p>
<p>Note that this is an <i>either-or</i> constraint, 
not simultaneous.  To compute to a specified absolute error, set <em>epsrel</em> to zero.  To compute to a specified relative error,
set <em>epsabs</em> to zero.  
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>
<p>The algorithms in <small>QUADPACK</small> use a naming convention based on the
following letters,
</p>
<div class="display">
<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></div>

<p>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.
</p>
<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top">&bull; <a href="Integrands-without-weight-functions.html#Integrands-without-weight-functions" accesskey="1">Integrands without weight functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Integrands-with-weight-functions.html#Integrands-with-weight-functions" accesskey="2">Integrands with weight functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Integrands-with-singular-weight-functions.html#Integrands-with-singular-weight-functions" accesskey="3">Integrands with singular weight functions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
</table>

<hr>
<div class="header">
<p>
Next: <a href="QNG-non_002dadaptive-Gauss_002dKronrod-integration.html#QNG-non_002dadaptive-Gauss_002dKronrod-integration" accesskey="n" rel="next">QNG non-adaptive Gauss-Kronrod integration</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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