Wave Propagation in Linear Media | re-examined

7.5 Test of the quadrature routine
7.5 Test of the quadrature routine
We now want to apply our quadrature routine to some of the test functions that are commonly
used in the literature. The objective of such benchmarks usually is to examine how
many evaluations of the integrand function a given algorithm needs to return a result of the
required accuracy. The number of function evaluations is in turn a measure for the e ectiveness
of competing algorithms. In the case of our quadrature routine, however, we obtain no
information how often Mathematica needs to evaluate the integrand for its internal computations.
All we can test is the accuracy of the result depending on the settings of the algorithm
parameters. So the main goal of this section is to verify that the algorithm works correctly
rather than to compare it with other routines.
As the execution time of the routine is not our main concern here, we do not use explicit
enumeration functions that de ne the partition points for the quadrature. We just specify
the argument function of the oscillating factors of the integrand and leave the determination
of the appropriate subdivision of the integration range to Mathematica.
Remark (Pure functions) To avoid unnecessary declarations, we use pure functions
[140] to specify the functions that are needed to compute the partition. In such a function
the places where the independent variables are to be inserted upon evaluation are marked
with slots (#, ##, #1, ::: ), the function itself is de ned as pure by a trailing ampersand (&).
So while we would de ne a quadratic polynomial in full notation as p[x_] := x^2 + 3 x,
we write the pure function as (#^2 + 3 #)&. This concept enables us to write very
compact code, but it is useful only if the function is not required to have a name.
Prior to testing we must load the package.
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