A necessary condition for the convergence of the accelerated overrelaxation (AOR) method (Q1122937)

In the application of the AOR method to the system \(Ax=b\) appears the matrix \(L_{r,\omega}=I-\omega (I-rL)^{-1}A,\) (L strictly lower triangular part of A; r,\(\omega\) real parameters). The author obtains, for \(r\neq 0\), the necessary condition of convergence: \(| \omega (1- r)| <| \omega -r| +| r|,\) using the relationship between the eigenvalues of \(L_{r,\omega}\) and of \(L_{r,r}=L_ r(=SOR\) matrix). Two equivalent statements are stated. For \(r=0\) the condition is \(0<\omega <2\).