  This paper deals with speeding up the convergence of a class of two-step
iterative methods for solving linear systems of equations. To implement the
acceleration technique, the residual norm associated with computed
approximations for each sub-iterate is minimized over a certain two-dimensional
subspace. Convergence properties of the proposed method are studied in detail.
The approach is further developed to solve (regularized) normal equations
arising from the discretization of ill-posed problems. The results of numerical
experiments are reported to illustrate the performance of exact and inexact
variants of the method on several test problems from different application
areas.
