On a class of finite step iterative methods (Conjugate directions) for the solution of an operator equation arising in electromagnetics

Abstract

A class of finite step iterative methods for the solution of linear operator equations is presented. Specifically, the basic principles of the method of conjugate directions are developed. Gaussian elimination and the method of conjugate gradients are then presented as two special cases. With an arbitrary initial guess, the method of conjugate gradient always converges to the solution in at most <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> iterations, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> is the number of independent eigenvalues for the operator in the finite dimensional space in which the problem is being solved. The conjugate gradient method requires much less storage ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sim 5N</tex> ) than the conventional matrix methods ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sim N^{2}</tex> ) in the solution of problems of higher complexity. Also, after each iteration the quality of the solution is known in the conjugate gradient method. The conjugate gradient method is also superior to the spectral iterative method as the latter does not always converge and it doubles the complexity of a given problem, unnecessarily. Four versions of the conjugate gradient method are presented in detail, and numerical results for a thin wire scatterer are given to illustrate various properties of each version.