Wave Propagation in Linear Media | re-examined

7.3 Implementation of OscInt and related functions
in order to nd a rule that matches a given set of arguments. If the two de nitions were
interchanged, a list fb,b0g would match the pattern a , and consequently the variable a
would erroneously be assigned the value fb,b0g.
The function rst extracts the values of a couple of relevant options and then checks whether
error control is required. If both AccuracyControl and PrecisionControl equal Infinity,
then no error control is used, and OscInt performs a straightforward computation. The lower
limit of the list of subdivision points is set either to a or maxfa; a0g, the latter being calculated
numerically at a higher precision, because the function Max works only with numbers but not
with symbols.
Remark (Options and defaults) The ltering of the options is worth a separate
mention [146]. It is accomplished by a double replacement rule like for instance in wp
= WorkingPrecision/.foptsg/.Options[OscInt], which works as follows: If the user
speci es a value WorkingPrecision->prec for this option, then it is passed to the function
in the argument opts. The statement is now evaluated from the left to the right, and
the rst replacement evaluates to prec. The second substitution fails because the list
of default options obtained by Options[OscInt] contains of course no rule prec->x. If,
on the other hand, the function is called without any option, then the rst replacement
cannot be evaluated. In this case, the second rule matches and wp is set to the default
value.
The number of partition points is obtained from NSumTerms plus NSumExtraTerms plus one,
so that the sequence of partial sums has the correct length for extrapolation. With the list
of subdivision points, PartInt is called to compute the actual integral.
If error control is to be used, a reasonable value for the WorkingPrecision is calculated
from the machine precision plus one, the parameter wp that might have been set by the user
(or equals the default value otherwise), and the PrecisionControl parameter if applicable.
This working precision, which might di er from the user's speci cation, is then used in the
call of the function OscIntControlled. Note that this particular option is placed before
the other options opts, which causes the new setting for WorkingPrecision to override any
other speci cation for this option, because Mathematica takes the rst value it encounters
and disregards all subsequent ones.
7.3.2 PartitionTable
This function actually only calls PartitionOffs and PartitionPoints, but is nonetheless
useful for the computation of a list of subdivision points. Its syntax is
PartitionTable[zerof, a, n, opts]
with the same input arguments as OscInt except for n, which denotes the number of partition
points to be computed. The options are the same as for OscInt.
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