Semiconvergence of extrapolated iterative method for singular linear systems
To solve the consistent linear system \(Ax=b\), the extrapolated iterative method \(x^{k+1}=(1-\omega)x^k + \omega(Tx^k+c)\) with \(A=M-N\), \(c=M^{-1}b\), \(T=M^{-1}N\) is considered in the case that \(A\) is singular. The main theorem states that convergence occurs when \(I-T\) has index 1, and one of the following two conditions are satisfied: either \({\text Re}(\lambda)<1\) for all \(\lambda\in S\) and \(0<\omega<\min_{\lambda\in S} E(\lambda)\) or \({\text Re}(\lambda)>1\) for all \(\lambda\in S\) and \(0>\omega>\max_{\lambda\in S} E(\lambda)\) where \(S=\sigma(T)\setminus\{1\}\) and \(E(\lambda)=2(1-{\text Re}\lambda)/(1-2{\text Re}\lambda+| \lambda| ^2)\). This generalizes results of the author for the nonsingular case [Appl. Numer. Math. 27, No. 3, 203--209 (1998; Zbl 0927.65052)] and of \textit{Y. Song} for the singular case [J. Comput. Appl. Math. 106, No. 1, 117--129 (1999; Zbl 0930.65033)].
scientific article; zbMATH DE number 2116555