Modification Methods for Inverting Matrices and Solving Systems of Linear Algebraic Equations

Abstract

Abstract.Modification methods for inverting matrices and solving systems of linear algebraic equations are developed from Broyden's rank-one modification formula. Several algorithms are presented that take as few, or nearly as few, arithmetic operations as Gaussian elimination and are well suited for the handling of data. The effect of rounding errors is discussed briefly.Some of these algorithms are essentially equivalent to, or "compact" forms of, such known methods as Sherman and Morrison's modification method, Hestenes' biorthogonalization method, Gauss-Jordan elimination, Aitken's below-the-line elimination method, Purcell's vector method, and its equivalent, Pietrzykowski's projection method, and the bordering method. These methods are thus shown to be directly related to each other.Iterative methods and methods for inverting symmetric matrices are also given, as are the results of some computational experiments. Introduction.This paper is concerned with the related problems of inverting matrices and solving linear nonhomogeneous algebraic systems of equations by a class of direct methods which we shall call modification methods. The adjective modification is used to describe these methods, for they are all based upon the modification of a matrix and its inverse by a matrix of rank one. Most of these methods are direct-that is, a solution to the problem is obtained by using a finite number of elementary arithmetic operations. The general formula that these methods are based upon, however, is iterative in nature, and iterative methods are also given.Modification