Variations on Richardson's method and acceleration

The aim of this paper is to present an acceleration procedure based on projection and preconditioning for iterative methods for solving systems of linear equations. A cycling strategy leads to a new iterative method. These procedures are closely related to Richardson’s method and acceleration. Numerical examples illustrate the purpose. For solving the system of linear equations Ax = b, we consider any convergent method which produces the sequence of iterates (xn). Quite often the convergence is too slow and it has to be accelerated. There exist many processes for that purpose whose references can be easily found. They are either quite general extrapolation methods as described, for example, in [5] or particular ones as those studied in [2]. The aim of this paper is to present a new acceleration procedure based on projection and preconditioning. Cycling with this procedure will lead to a new iterative method. 1 The procedure Let us consider the sequence (yn) given by yn = xn − λnzn (1) where zn is an (almost) arbitrary vector called the search direction or the direction of descent and λn a parameter called the stepsize. Setting rn = b − Axn and ρn = b− Ayn, we have ρn = rn + λnAzn. (2) 1991 Mathematics Subject Classification : 65F10, 65F25, 65B05, 65D15.