Loss and retention of accuracy in affine scaling methods

Abstract

This paper considers affine scaling methods for solving linear l1 problems of the form The paper explains the motivation behind the method and describes its basic steps. Analysis of the method reveals several interesting features that characterize its asymptotic behaviour. Special attention is given to propagation of rounding errors. It is shown that the formal affine scaling algorithm has an inherent drawback: At the kth iterationk= 1,2,..., the algorithm solves a linear least squares problem and uses its residual vector, rk, to compute a search direction uk. However, as kincreases tends to zero, while the rounding errors in ukgrow faster than where e denotes the machine precision in our computations. That is, the smaller the larger the error. Thus, although in theory in practice ukcontains an error vector that belongs to Range(A). This error component is the major reason for accumulation of rounding errors. Cancelling this component avoids most of the damage. The paper explains how to overcome this difficulty. Numerical experiments illustrate the deteriorating effects of small residuals and the usefulness of the proposed safeguards.