Wave Propagation in Linear Media | re-examined

7 Mathematica implementation of a quadrature function
WorkingPrecision is the standard option to set the precision of the internal calculations,
with $MachinePrecision as default value. Normally it may be set by the user
to any arbitrary value (between the bounds $MinPrecision and $MaxPrecision, of
course). If accuracy or precision control are used, then its treatment is slightly different,
because these functions need high-precision arithmetic which in turn requires a
working precision that is higher than the machine precision. Thus WorkingPrecision
is set to the maximum of $MachinePrecision plus one digit, the precision probably
speci ed by the user, and the value of PrecisionControl plus 13 digits | provided
that precision control is used at all. The value of AccuracyControl cannot be used to
set the working precision since this would require one to know the order of magnitude
of the integral beforehand.
FunctionType 2 fSinArgument, CosArgument, ZeroListg selects how the function
fzero is to be interpreted. ZeroList means that the function returns consecutive
partition points. The other possibilities specify the circular function whose zeros are
computed for the use as subdivision points. For SinArgument, which is also the default
value, the equation (x) = k is solved, for CosArgument, it is the equation
(x) = k + =2. If the oscillating factor u( ) is a pure circular function, then this
option additionally allows one to choose the extrema rather than the zeros as partition
points.
The default options of OscInt cause the function to run at machine precision and with the
control loop for the precision of the result disabled. The second argument is in this case
regarded as the argument (x) of the oscillation. These default settings make up the fastest
and most user-friendly con guration.
7.2 Structure of the package
The function OscInt is actually nothing but a driver that calls other functions depending on
the settings of various options. Fig. 7.1 shows the respective functions and their dependencies.
The most important distinction inside the package is that between the use of error control
for the extrapolation and the plain computation of the integral without any feedback loop.
Depending on this decision, OscInt invokes the following functions:
PartitionTable computes a list of subdivision points with a given length, starting at
a suitable point above the lower integration limit. In fact it only calls the functions
PartitionOffs and PartitionPoints with a reduced set of input parameters.
PartInt takes as input a set of subdivision points and computes the integral by extrapolation.
Like PartitionTable, itis used only for straightforward quadrature without
error control.
OscIntControlled is the version of OscInt used with error control. It comprises a
control loop that monitors the precision of the extrapolation result and increases the
length of the sequence of partial sums if needed.
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