The extrapolated first order method for solving systems with complex eigenvalues (Q760158)

The extrapolation scheme \(x^{(m+1)}=(\alpha S+(1- \alpha)I)x^{(m)}+\alpha c=:S_{\alpha}x^{(m)}+\alpha c\) for the solution of a linear system in fixed point form \(x=Sx+c\) is considered. Under the assumption that the eigenvalues of S are contained in some rectangle R, bounds for \(\alpha\) in terms of parameters describing R are derived such that \(S_{\alpha}\) is convergent. Further ''good'' (and sometimes optimal) values of \(\alpha\) are determined. Similar results have been obtained by \textit{A. Yeyios} [Linear Algebra Appl. 57, 191-203 (1984; Zbl 0527.65027)].