Methods for Acceleration of Convergence (Extrapolation) of Vector Sequences

An important problem that arises in different areas of science and engineering is that of computing limits of sequences of vectors fxmg, where xm 2C, and the dimension N is very large in many applications. Such vector sequences arise, for example, in the numerical solution of very large systems of linear or nonlinear equations by fixedpoint iterative methods, and limm!1xm are simply the required solutions to these systems. One common source of such systems is the finite-difference or finite-element discretization of continuum problems. In most cases, however, the sequences fxmg converge to their limits extremely slowly. That is, s 1⁄4 limm!1xm can be approximated with a prescribed level of accuracy by xm with very large m. Clearly, this way of approximating s via the xm becomes very expensive computationally. One practical way of tackling this problem effectively is by applying to the sequence fxmg a suitable convergence acceleration method (or extrapolation method). More specifically, let us consider the (linear or nonlinear) system of equations