Wave Propagation in Linear Media | re-examined

7.3.4 PartitionPoints
7 Mathematica implementation of a quadrature function
This function returns a list of subdivision points in ascending order. The way to call it is
PartitionPoints[zerof, a, n, offs, irel, opts],
zerof being the well-known function (x) orx (n). The argument a denotes the lower bound
of the sequence, n is the desired number of points. The variable offs is the o set o obtained
from PartitionOffs. This function could have been merged into PartitionPoints (and in
fact had been in a former version of PartitionTable), but as for instance OscIntControlled
makes multiple calls to PartitionPoints to gradually build up the table of subdivision points,
is was deemed more e cient to split these two functional blocks. The parameter irel denotes
the index r 0 of the rst point and can be used to compute subdivision tables in two or
more portions.
Like PartitionOffs, this function also behaves di erently for di erent values of the option
FunctionType. The simplest case is when zerof returns the partition points themselves
(FunctionType->ZeroList). Then we just need to build up the table fx (r);x (r+
1);::: ;x (r+n,1)g.
Things are more complicated when (x) is given directly (FunctionType->SinArgument or
CosArgument). Then the evaluation depends on the behaviour of (x) at x ! 1. If the
limit is de nite, then the integrand is not in nitely oscillating, and a partition extrapolation
quadrature is not applicable at all. In this case, we return Infinity so that the rst partial
integral already covers the entire range. For an inde nite limit, we must distinguish between
increasing and decreasing functions and nally obtain the partition point from the following
equations:
zi :
8
><
>:
xi = 1 if ,1Infinity]]],
Table[Infinity,fk,0,n-1g],
N[zerof'[offs+1]] > 0,
koffs = N[Ceiling[zerof[offs]/Pi],2 wp];
Re[Table[x/.FindRoot[zerof[x] == (k+koffs+cfoffs) Pi,
fx,offs,offs+1g],
fk,irel,irel+n-1g]],
True,
170
(7.2)