Comments on "On the use of /spl rho/-algorithm in series acceleration"
V.P. Pyati · IEEE Transactions on Antennas and Propagation · 1996
The author draws the attention of the readers to the paper of Singh and Singh (see ibid., vol.39, p.1514-16, Oct. 1991) that an extremely simple and elegant method of handling this age-old problem may be found in Morse and Feshbach (1953). Suggested by the geometry of the original problem for the Helmholtz equation, one first determines the solution of the Poisson's equation (setting the wavenumber to zero) with appropriate boundary conditions and sources in closed form, as well as an infinite series. A useful feature of the series solution is that each term resembles a corresponding term in the original series solution of the Helmholtz equation, and for large values of the summation index, term by term, equality is approached. By adding the closed form answer, subtracting the series solution, and rearranging, one realizes fast convergence. Yet another gratifying feature is that convergence of the transformed series can be easily proven.