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