Computing 33, 349 - 352 (1984) Computing © by Springer-Verlag 1984 Short Communications / Kurze Mitteilungen A Note on the ART of Relaxation M. R. Trummer, Ziirich Received February 28, 1983 Abstract - Zusammenfassung A Note on the ART of Relaxation. The ART algorithm, an iterative technique for solving large systems of linear equations, is shown to converge even for inconsistent systems, provided the relaxation parameters are chosen appropriately. The limit is a weighted least squares solution. AMS Subject Classifications: 65FI0, 15A09, 92A07. Key words: Image reconstruction, linear equations, iterative methods, generalized inverses. Der ART -Algorithmus mit Unterrelaxation. Die Konvergenz des ART -Algorithmus, ein iteratives Verfahren zur Lasung linearer Gleichungssysteme, wird bewiesen. Bei geeigneter Wahl der Relaxationsparameter konvergiert der Algorithmus selbst im Faile inkonsistenter Systeme, und zwar gegen eine Kleinste-Quadrate-Lasung . . 1. Introduction This note deals with strong underrelaxation in the ART algorithm, introduced in [3J, for inconsistent systems. This method for solving linear equations iteratively has already been suggested by Kaczmarz [5J in 1937; it is e.g. used in image reconstruction where huge, sparse systems have to be solved. Let be a m by n matrix; we want to solve the system Ax=y. The linear ART algorithm with relaxation is defined by (1) (2) (3) (4) x (0) EAT (lRm) x(km+j):=x(km+j-1)+rk II aj 11-2 [Yj-aJ x(km+j-1)J aj, 1 ~j ~m, k=O, 1,2, .... The rk are called relaxation parameters. This algorithm is a row action method [lJ, i. e. an iterative procedure using one row of A at each step. It is known [8J that under