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<A NAME=num_int>

<TITLE>num_int</TITLE></A>
<b><a href=r37_idx.html>INDEX</a></b><p><p>



<B>NUM_INT</B> _ _ _  _ _ _  _ _ _  _ _ _ <B>operator</B><P>
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For the numerical evaluation of univariate integrals 
over a finite interval the following strategy is used: 
If 
<A HREF=r37_0154.html>int</A> finds a formal antiderivative 
 which is bounded in the integration interval, this 
 is evaluated and the end points and the difference 
 is returned. 
Otherwise a 
<A HREF=r37_0430.html>Chebyshev fit</A> is computed, 
 starting with order 20, eventually up to order 80. 
 If that is recognized as sufficiently convergent 
 it is used for computing the integral by directly 
 integrating the coefficient sequence. 
If none of these methods is successful, an 
 adaptive multilevel quadrature algorithm is used. 
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<P>
For multivariate integrals only the adaptive quadrature is used. 
This algorithm tolerates isolated singularities. 
The value <em>iterations</em> here limits the number of 
local interval intersection levels. 
&lt;a&gt; is a measure for the relative total discretization 
error (comparison of order 1 and order 2 approximations). 
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 <P> <H3> 
syntax: </H3>
<em>num_int</em>(&lt;exp&gt;,&lt;var&gt;=(&lt;l&gt; .. &lt;u&gt;) 
 [,&lt;var&gt;=(&lt;l&gt; .. &lt;u&gt;),...] 
 [,accuracy=&lt;a&gt;][,iterations=&lt;i&gt;]) 
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<P>
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where &lt;exp&gt; is the function to be integrated, 
&lt;var&gt; are the integration variables, 
&lt;l&gt; are the lower bounds, 
&lt;u&gt; are the upper bounds. 
 <P>
<P>
Result is the value of the integral. 
 <P>
<P>
 <P> <H3> 
examples: </H3>
<P><PRE><TT>
num_int(sin x,x=(0 .. 3.1415926));

  2.0000010334

</TT></PRE><P>