Wave Propagation in Linear Media | re-examined

7.3.3 PartitionOffs
7 Mathematica implementation of a quadrature function
The aim of the function PartitionOffs is to nd an index o set for the subsequent computation
of the partition points such that we can assume that the point with index zero is the
smallest possible one | irrespective of the lower integration limit or any particular shape of
the argument function (x). The function call is
PartitionOffs[zerof, a, opts],
with zerof as function de ning either the argument (x) oranenumerating function for the
partition points x (n). The parameter a is not necessarily the lower integration limit but
can also be any arbitrarily chosen abscissa point meant asalower bound for the sequence of
partition points. The options are the same as for OscInt.
How the function is evaluated chie y depends on the option FunctionType that speci es how
zerof is to be interpreted. We shall rst discuss the case where FunctionType->ZeroList,
which means that zerof directly returns the subdivision points x (n); n 0. This function
has to be de ned following one important rule: the rst point it returns must be larger than
the position of any extremum of (x) or its derivatives, so both (x) and its derivatives
must be monotonic for x x (0). Then we must simply nd the smallest index o 0
where x (o) a. This operation is performed by the following statement, where the equation
x (na) =ais solved numerically and the result is set to o = dnae. For this purpose, zerof[n]
must be able to accept input parameters of type real although the value it returns makes sense
only for integer input values. The option MaxIterations is set to the value of the OscInt
option MaxRecursion, AccuracyGoal in this particular case equals to PartitionAccuracy.
If[N[zerof[0]] < N[a],
Ceiling[x/.FindRoot[zerof[x] == a,fx,0,1g,
WorkingPrecision->wp,
AccuracyGoal->pa,
MaxIterations->ma]],
0]
Remark (Implementation) To determine whether x (0)