Weighted Averages and Double Exponential Algorithms

Abstract

SUMMARYThis paper reviews two simple numerical algorithms particularly useful in Computational ElectroMagnetics (CEM): the Weighted Averages (WA) algorithm and the Double Exponential (DE) quadrature. After a short historical introduction and an elementary description of the mathematical procedures underlying both techniques, they are applied to the evaluation of Sommerfeld integrals, where WA and DE combine together to provide a numerical tool of unprecedented quality. It is also shown that both algorithms have a much wider range of applications. A generalization of the WA algorithm, able to cope with integrands including products of Bessel and similar oscillatory functions, is described. Similarly, the original DE algorithm is adapted with exceptional results to the evaluation of the multidimensional singular integrals arising in the discretization of Integral-Equation based CEM formulations. The new possibilities of WA and DE algorithms are demonstrated through several practical numerical examples.

The Sommerfeld half-space problem revisited: from radio frequencies and Zenneck waves to visible light and Fano modes

“…If the upper half-space is lossless (air), the branch point k 1 lies on the real axis, and the SIP must be indented above it. Although effective methods for the computation of the Sommerfeld integrals along the SIP are available, [83][84][85][86][87] the rapid oscillations of the Bessel functions on the real axis make them inefficient for large ρ. In this case, more efficient integral representations can be obtained by a suitable deformation of the integration path and analytic continuation of integrand functions into the complex k ρ -plane.…”

Nonlinear filtering using the double exponential transformation

“…27 The method has also been extended to multiple integration 28 and in the computation of multivariate integrals with endpoint singularities. [29][30][31] A comparison of the distribution of sample points for DE quadrature and Gauss-Legendre quadrature over the interval [−1, 1] is shown in Fig. 1.…”