Wave Propagation in Linear Media | re-examined

6.2 Choosing the rst partition point
0.12
0.1
0.08
0.06
0.04
0.02
20 40 60 80 100
The surprising step in the sequence stems from the minimum of the argument function that
causes the distance of successive zeros to be enlarged in its neighbourhood, which in turn
results in exceedingly large partial integrals. For the rst 50 elements, this e ect is hidden
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by the dominating factor in the integrand and the sequence seems to converge. Indeed,
x+1
the extrapolation algorithm returns a valid limit without any warning when we feed it with
the sequence elements 20 to 30.
In[9]:= SequenceLimit[Take[partial,f20,30g]]
Out[9]= 0.0313787
We get a result even if we use the portion of the sequence that seems to diverge as we approach
the extremum of the argument function.
In[10]:= SequenceLimit[Take[partial,f60,70g]]
Out[10]= 0.0313787
Only if we start the extrapolation behind the step, we obtain the correct limit and the real
value of the integral.
In[11]:= SequenceLimit[Take[partial,f80,90g]]
Out[11]= 0.109863
Remark (Possible warnings) If we had replaced the factor x+1 by x2 , we would
+1
have received at least a warning that the algorithm suspected the results to be incorrect in
the rst two cases (the returned values would have been `correct' anyhow). However, one
cannot rely completely on such messages as there are other integrands where the results
are de nitely right despite these warnings.
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