A review is presented of the extrapolation methods for accelerating the convergence of Sommerfeld-type integrals (i.e. semi-infinite range integrals with Bessel function kernels), which arise in problems involving antennas or scatterers embedded in planar multilayered media. Attention is limited to partition-extrapolation procedures in which the Sommerfeld integral is evaluated as a sum of a series of partial integrals over finite subintervals and is accelerated by an extrapolation method applied over the real-axis tail segment (/spl alpha/,/spl infin/) of the integration path, where /spl alpha/>0 is selected to ensure that the integrand is well behaved. An analytical form of the asymptotic truncation error (or the remainder), which characterizes the convergence properties of the sequence of partial sums and serves as a basis for some of the most efficient extrapolation methods, is derived. Several extrapolation algorithms deemed to be the most suitable for the Sommerfeld integrals are described and their performance is compared. It is demonstrated that the performance of these methods is strongly affected by the horizontal displacement of the source and field points /spl rho/ and by the choice of the subinterval break points. Furthermore, it is found that some well-known extrapolation techniques may fail for a number of values of /spl rho/ and ways to remedy this are suggested. Finally, the most effective extrapolation methods for accelerating Sommerfeld integral tails are recommended.

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Extrapolation methods for Sommerfeld integral tails

The N-point Gaussian quadrature method is generalised to cater for various possible singular behaviours at the end points of the interval of integration at the expense of being algebraically exact for a polynomial of lower order than usual Weights and abscissae are chosen to exactly integrate an integrand which is the sum of the singular functions and an arbitrary polynomial This allows us to cater for several different end-point singularities in the same quadrature formula and in this way differs from published quadratures where a singular behaviour is incorporated in a weight function that multiplies an arbitrary polynomial We present tables of weights and abscissae that cater for (i) logarithmic end point singularities and (ii) logarithmic plus inverse square root singular behaviours Also a 10-point quadrature is presented that exactly caters for log(x), x-1/4,x-1/2,x-3/4 singular behaviours and is recommended for programmable calculator use Finally a brief comparison study of the various (10-poin

Extension of numerical quadrature formulae to cater for end point singular behaviours over finite intervals

On numerical computation of integrals with integrands of the form f(x)sin(w/xr) on [0, 1]

Computation of potentials from current electrodes in cylindrically stratified media

An integration technique constructed to treat integrands with singularity has been extended to the solutions of the equation of radiative transfer in integral form. To illustrate the technique, radiative transfer is considered in an absorbing and isotropic scattering solid cylinder exposed to diffuse radiation, as well as radiative equilibrium in a participating medium between concentric cylinders with unequal surface temperatures. The integral equations for the two problems are solved by using the collocation method. Since the integral equations have singular kernels, the partition-extrapolation technique is employed. The technique yields estimates of the integration with singularity by extrapolation of the integration results of several subdomains without singularity. Comparisons of the results obtained by the present method with those obtained by the F-N method and the Monte Carlo method show that the partition-extrapolation technique generates accurate results.

Partition-extrapolation integration applied to radiative transfer in cylindrical media.

Partition-extrapolation methods for numerical quadratures are based on dividing the range of integration into an infinite number of subdomains, evaluating the subintegrals over a finite number of subdomains numerically, and then estimating the integral by extrapolation of the partial sums. There is considerable flexibility in the choice of subdivisions, method of evaluating the subintegrals and extrapolation procedure. The treatment is heuristic, it being shown that the present methods compare favorably with previous methods for a variety of examples involving semi-infinite ranges or integrands with singularities.

Partition-extrapolation methods for numerical quadratures

The results obtained for a set of integrals with singularities at the endpoints using the Harris and Evans modified ten points Gauss formula are compared with those using Stenger's formula. While the Gaussian approach gives better results for a comparable number of points, the Stenger rule is easily applied with an arbitrary number of nodes and the comparison should be more favorable when more nodes are used.

A comment on quadrature in the presence of end point singularities

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Partition-extrapolation methods for numerical quadratures

A comment on quadrature in the presence of end point singularities