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<a name="Integrands-without-weight-functions"></a>
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Next: <a href="Integrands-with-weight-functions.html#Integrands-with-weight-functions" accesskey="n" rel="next">Integrands with weight functions</a>, Up: <a href="Numerical-Integration-Introduction.html#Numerical-Integration-Introduction" accesskey="u" rel="up">Numerical Integration Introduction</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Integrands-without-weight-functions-1"></a>
<h4 class="subsection">17.1.1 Integrands without weight functions</h4>
<a name="index-Gauss_002dKronrod-quadrature"></a>
<p>The algorithms for general functions (without a weight function) are
based on Gauss-Kronrod rules. 
</p>
<p>A Gauss-Kronrod rule begins with a classical Gaussian quadrature rule of
order <em>m</em>.  This is extended with additional points between each of
the abscissae to give a higher order Kronrod rule of order <em>2m+1</em>.
The Kronrod rule is efficient because it reuses existing function
evaluations from the Gaussian rule.  
</p>
<p>The higher order Kronrod rule is used as the best approximation to the
integral, and the difference between the two rules is used as an
estimate of the error in the approximation.
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