The Application of the Conjugate Gradient Method for the Solution of Operator Equations in Electromagnetics

The objective of this paper is to survey many of the popular methods utilized in solving numerical problems arising in electromagnetics. Historically, the matrix methods have been quite popular. One of the primary objectives of this paper is to introduce a new class of iterative methods, which have advantages over the classical matrix methods in the sense that a given problem may be solved to a prespecified degree of accuracy. Also, these iterative methods (particularly conjugate gradient methods) converge to the solution in a finite number of steps irrespective of the initial starting guess. Numerical examples have been presented to illustrate the principles. The conjugate gradient method solves a Toplitz system in θ(N) steps when applied in conjunction with Fast Fourier Transform. The conjugate gradient method essentially performs a singular value decomposition for ill-conditioned systems. This is illustrated by the solution of the deconvolution problem.

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