Wave Propagation in Linear Media | re-examined

8.3 Implementation of the quadrature modules
quadrature. There are of course other function declarations necessary for the other regions
(incident and re ected waves). However, they look very much the same as the mentionend
integrands and thus are not given here.
The quadrature modules consist of two parts, the computation of the parameters required for
the function calls and the quadrature itself. Note that the parameters b and d are evaluated
numerically. In the case of b this is necessary to ensure the correct operation of the functions
QuadOffset and QuadZero that comprise conditional statements and are not prepared to
take symbolical arguments. On the other hand, it improves the performance of the module
because it reduces the computation time by up to ten percent (depending on the actual
parameter values). For the same reason, the o set parameter of the function enumerating the
zeros is calculated before the quadrature starts. To prevent inaccuracies due to the numerical
evaluation of the parameters, they are calculated to a precision two digits higher than the
WorkingPrecision.
Remark (Version peculiarities) In the original version of the package the o set values
were not computed numerically as shown in the example given below. The function
QuadOffset thus returned results like Ceiling[const/Pi] that were left unevaluated
because Mathematica treats Pi as a symbol rather than as a number. This worked
well most of the time, however, for particular sets of input parameters the quadrature
function would never terminate. This peculiar behaviour seemingly depended on the
software version (2.2) but not on the platform, which was veri ed on PCs as well as on
SUN and HP workstations. Neither version 2.1 nor the new version 3.0 caused similar
troubles. The problem could be overcome only by changing the results of QuadOffset
into numbers, which in turn provides another slight performance improvement.
The quadrature function OscInt is called four times with the arguments of the integrands set
to the appropriate values taken from (8.30). In this connection it is important to notice that
owing to the uniform structure of the integrand functions we need to pass only the name of the
function to the template, the arguments are then added inside the module. Therefore we can
use the template for all parts of the wave. The results of the individual function calls are then
summed up with the coe cients given in (8.30) and multiplied with the waveform-dependent
constant.
RectTransTemp[fpos_, fneg_, x_, t_, w_, k_, opts___Rule] :=
Module[fa,b,c,d,o11,o12,o21,o22,
wp = WorkingPrecision/.foptsg/.Options[OscInt]g,
a = -t;
b = N[2 Pi k/Sqrt[w],wp+2];
c = x;
d = N[-2 Pi k,wp+2];
o11 = N[QuadOffset[a, c,0],wp+2];
o12 = N[QuadOffset[a,-c,0],wp+2];
o21 = N[QuadOffset[a, b+c,d],wp+2];
o22 = N[QuadOffset[a,-b-c,d],wp+2];
(-(OscInt[fpos[a, 0, c,0,w,k],
QuadZero[a, c,0,o11],1,FunctionType->ZeroList,opts] +
201