IMPROVING THE SOLVABILITY OF ILL-CONDITIONED SYSTEMS OF LINEAR EQUATIONS BY REDUCING THE CONDITION NUMBER OF THEIR MATRICES

Abstract

This paper is concerned with the solution of ill-conditioned Systems of Linear Equations (SLE's) via the solution of equivalent SLE's which are well-conditioned. A matrix is rst constructed from that of the given ill-conditioned system. Then, an adequate right-hand side is computed to make up the instance of an equivalent system. Formulae and algorithms for computing an instance of this equivalent SLE and solving it will be given and illustrated. Ill-conditioned systems of linear equations (SLE's) are notoriously difficult to solve to any useful accuracy, (5). Their matrices are characterized by large condition numbers. The difficulty may be negotiated by solving different but equivalent systems which are well-conditioned well-conditioned SLE's have matrices with small condition numbers. The approach we put forward here constructs a new matrix and a new right-hand side that constitute an instance of an equivalent SLE to the one given which is ill-conditioned. Moreover, this new matrix has a small condition number compared to that of the matrix of the initial SLE. This means that solving this equivalent system must be better than solving the original one by virtue of the difference in the magnitude of the condition numbers of their matrices. Before embarking in the derivation of the necessary formulae and algorithms to construct such equivalent SLE's, let usrst motivate this approach. Consider the following system of linear equations (SLE) in two variables.