Über Nullräume, Wertebereiche und Relationen von Operatoren, die bei instationären IterationsverIahren zur Lösung linearer Gleichungen auftreten

Abstract

Iteration methods of the form

are considered to obtain solutions or generalized solutions of the linear equation $Ax=b$. Thereby the residues $r_n=b-A{x}_n$ satisfy the iteration procedure

The existence of $B_{\infty }={\lim }_{n\to \infty }{B}_n$ already guarantees the existence of the remaining limits $P_{\infty },Q_{\infty },x_{\infty }$ and $r_{\infty }$. Conditions are given, under which $B_{\infty }$ exists and possesses certain additional properties. These conditions are expressed on the one hand by operator equations and on the other hand by relations between null spaces and ranges of operators. The results are applied to two classes of iteration methods.