Equivalent conditions for the convergence of extrapolated method for linear systems

We further investigate the convergence of extrapolated method for solving linear systems. Combined with the results in Song and Wang [Y. Song, L. Wang, On the semiconvergence of extrapolated iterative methods for singular linear systems, Appl. Numer. Math. 44 (2003) 401-413], equivalent conditions are obtained. The new conditions are based on the field of values of normal matrices which are easier to estimate. Some Numerical examples are given to illustrate our results.