Wave Propagation in Linear Media | re-examined

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6Towards a quadrature routine
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Figure 6.1: Sequences of partial sums for R 1 sin x
0 x dx. The partition points are the zeros ( ) or extrema
( ) of the integrand. The inner sequences are obtained when the integration range for the subintervals
is a full period.
In[6]:= sequ = Table[NIntegrate[Sin[x]/x,
fx,2 i Pi + Pi/2,2 (i+1) Pi + Pi/2g],
fi,0,100g];
partial = FoldList[Plus,NIntegrate[Sin[x]/x,fx,0,Pi/2g],sequ];
SequenceLimit[partial] - N[Pi/2]
Out[6]= -7
-2.13857 10
Fig. 6.1 shows the sequences of partial sums calculated for this example and reveals that the
extrema are indeed more suitable as partition points because the errors jSn , Sj decrease like
1=n 2 in this case. When the zeros are used to subdivide the integration range, the errors
behave like 1=n, which gives only half the speed of convergence.
Proof We shall rst derive an estimate for the magnitude of the partial integrals. The second meanvalue
theorem states that if two functions f(x) and g(x) are continuous on [a; b] and g(x) is also
monotonic, then there exists a point 2 [a; b] such that
Z b
a
Z
f(x) g(x) dx = g(a+)
a
Z b
f(x) dx + g(b,) f(x) dx : (6.2)
We set f(x) = sin x, g(x) =1=x and choose the integration interval [(n + ) ; (n +1+ ) ], where
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