Wave Propagation in Linear Media | re-examined

6.2 Choosing the rst partition point
6Towards a quadrature routine
When using an extrapolation method to determine the value of an integral, we face the
problem that we must conclude from a nite sequence of partial sums to its limit. While it
is normally easy to judge the existence of an inde nite integral, it takes some considerations
to nd the right starting point for the sequence being subject to the extrapolation algorithm.
If the choice was wrong, the algorithm may in some cases return a warning message if it
encounters convergence problems. However, the checking capabilities are limited and strongly
data dependent, and most extrapolation algorithms even nd nite results for divergent series.
Hence the responsibility to be careful always rests with the user.
To illustrate the possible di culties, we regard a simple example. Consider the integral
I =
Z 1
a
1
x +1 sin (x , xm) 2 dx : (6.8)
We want toevaluate it through extrapolation and choose the partition points to be the zeros
of the sin-function. For the sake of simplicity, we assume that the lower integration limit a
be the smallest positive zero. As we need the sequence of partial sums, we must de ne a
function that returns all zeros of the integrand in ascending order, starting with index 0. To
that end, it is essential to distinguish the two branches of the parabolic argument (x,xm) 2 .
It is easy to see that there are o = bxm 2 = c zeros between the lower limit of integration and
the minimum of the argument function at x = xm, so this value can be used as an index o set
for the calculation of the zeros.
In Mathematica notation, the integrand and the function returning the k-th positive zero
xk are given below. The position of the extremum xm and the index o set o are global
parameters.
In[7]:= f[x_] := 1/(x+1) Sin[(x-xm)^2];
zero[k_] := If[k >= o, xm + Sqrt[(k-o) Pi],
xm - Sqrt[(o-k) Pi]];
Now we calculate the rst 100 elements of the sequence of partial sums with the zeros used
as subdivision points.
In[8]:= xm = 15;
o = N[Floor[xm^2/Pi]];
zmax = 100;
sequ = Table[NIntegrate[f[x],fx,zero[i],zero[i+1]g,
Method->GaussKronrod],fi,0,zmaxg];
partial = FoldList[Plus,0,sequ];
ListPlot[partial,PlotStyle->fPointSize[0.006]g];
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