Efficient solution of dense linear system of equations arising in the investigation of electromagnetic scattering by truncated periodic structures

Abstract

The problem of estimating the edge effects in a truncated periodic array is difficult because it requires the solution of a large, dense, linear systems of equations. In this paper we present an efficient numerical technique for solving the problem of one- and two-dimensional truncated array of scatterers. We utilize global impedance matrix compression, achieved by the reduced-rank representation of the off-diagonal blocks, together with partial-QR decomposition. The system with a compressed matrix is then solved by using an iterative method based on preconditioned transpose-free quasi minimal residual (PTFQMR) method, followed by further iterative refinements. Both the preconditioning and the compression steps are configured such that they can take advantage of the block structure of the matrix. A comparative evaluation of the performance of the present iterative technique is carried out vis-a-vis other matrix solution methods, both iterative and direct, with a view to demonstrating the superior computational efficiency of the present solver. We further show that a high matrix compression rate can be achieved without sacrificing the accuracy required by successive refinements. This not only leads to a saving in the memory requirements, and thus enables us to handle large problems which would otherwise be unmanageable, but also contributes to the numerical efficiency of the algorithm.