To solve \(Ax=b\), an iterative method \(x^{k+1}={\mathcal L}_{t,\omega}x^k+c\) may be defined using \[ {\mathcal L}_{t,\omega}=(I-\omega L)^{-1}[(1- t)I+(t-\omega)L+tU]. \] The asymptotic rate of convergence may be measured as \(R_{\infty}(X)=-\log (\rho (X))\) where \(\rho(x)\) is the spectral radius of matrix \(X\). Let \(B={\mathcal L}_{1,\infty}\), \(G={\mathcal L}_{1,1}\) and \(S={\mathcal L}_{\omega,\omega}\) be the Jacobi, Gauss-Seidel and SOR iteration matrices. The authors show that for an optimal choice of \(t\)
\[ \lim_{\rho (B)\to 1}-R_{\infty}({\mathcal L}_{t,1})/R_{\infty}(G)=2, \] and for an optimal choice of \(\omega\), \[ \lim_{\rho (B)\to 1}-R_{\infty}({\mathcal L}_{1,\omega})/[R_{\infty}(G)]^{1/2}=(2(p-1)/p)^{1/2}. \]
As the authors state, only the latter converges nearly as fast as the SOR method (when \(p\) is large) for which R. S. Varga [Matrix iterative analysis. Englewood Cliffs, New Jersey: Prentice-Hall (1963; Zbl 0133.08602)] has shown that \[ \lim_{\rho (B)\to 1}- R_{\infty}(S)/[R_{\infty}(G)]^{1/2}=(2p/(p-1))^{1/2}. \] The proofs utilize the concept of generally consistently ordered matrices. The final remark of the paper appears to refer to further work of the first two authors [The extrapolated Gauss-Seidel method plus semi-iterative method for generally consistently ordered matrices. Int. J. Comput. Math. 25, No. 1, 55–66 (1988; Zbl 0663.65027)].