Wave Propagation in Linear Media | re-examined

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6Towards a quadrature routine
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Figure 6.2: Computing time for a Gauss-Kronrod rule depending on the number of subintervals for
the integrand in (6.8) with xm = 40.
In[18]:= timtab = Table[fsteps,First[
Timing[
min = zero[0];
max = zero[510];
Sum[NIntegrate[f[x],fx,min + i (max-min)/steps,
min + (i+1) (max-min)/stepsg,
Method->GaussKronrod,MaxRecursion->15],
fi,0,steps-1g]
]
]/Secondg,fsteps,1,1401,10g];
Remark (Implementation) To speed up the computation, we evaluate the interval
bounds only once and save them as constants. The list is made up of the number of
intervals and the computing time for each trial, which is obtained by taking the rst
element of the list returned by the function Timing. To be able to plot this list, we
must rst divide each evaluation time by the constant Second. The actual results of the
individual computations are not stored separately as they are all correct.
The results in g. 6.2 show that with only a few subdivisions, the overall computing time
can be dramatically decreased. The minimum, however, is still above 30 seconds, which is
signi cantly higher than the values in tab. 6.1 . The equidistant partition is also the reason
why it takes about 800 subintervals to achieve the minimum although there are only 510
zeros in the integration range. The linear slope beyond the minimum stems from the time-
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