Direct Technique for Improving a Matrix Inverse

A method is shown that transforms the problem of inverting an ill-conditioned matrix to one of inverting a diagonally dominant matrix. An error analysis is outlined and the method is compared in theory and in result with the commonly used iterative methods. This direct method is demonstrated to be the limiting case of an nth-order iterative procedure as n approaches infinity. Examples are given that show the advantages of the direct method even under adverse conditions. The unreliability of the convergence of the iterative technique due to computational errors is also discussed. Introduction The problem of obtaining accurate inverses of large ill-conditioned matrices arises in many numerical procedures [ 1,2]. Application areas include structural analysis , electrical network analysis, and the solution of systems of differential equations with variable coefficients. To realistically evaluate any method, a detailed analysis of computational errors due to finite significance