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, ...
Computing
Computing
33, 349 - 352 (1984)
©
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, Key words: Image reconstruction,
15A09, 92A07.
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 (1)
be a m by n matrix; we want to solve the system Ax=y.
(2)
The linear ART algorithm with relaxation is defined by x (0) EAT x(km+j):=x(km+j-1)+rk
II
aj
(3)
(lRm) 11-2
[Yj-aJ
x(km+j-1)J
aj, (4)
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
M. R. Trummer:
350
certain conditions on the relaxation parametersrk the algorithm converges to the minimum norm solution of(2), provided a solution exists. However, for inconsistent systems, the sequence x (p) will not converge, unless rk -* O. On the other hand, the relaxation parameters must not tend to 0 too fast - otherwise, the limit 'would have nothing to do with a "solution". Since we are mostly concerned about inconsistent systems, we have to resort to the concept ofleast squares solutions: Let Pdenote the orthogonal projector onto A (IRn), and let A t denote the generalized inverse of A associated with P (see e. g. [4J). At y is the minimum norm solution of AT Ax = AT y, or equivalently, Ax=Py.
2. Convergence of ART Theorem 1: Suppose (a) (c) (b)
w k=O w
II
aj
II
= 1, 1 0 is investigated in [2].
2. Ifrk=r
References
[I] Censor, Y.: Row-action methods for huge and sparse systems and their applications. SIAM Review 23,444-446 (1981). [2] Censor, Y., Eggermont, P. P. B., Gordon, D.: Strong underrelaxation in Kaczmarz's method for inconsistent systems. Numer. Math. 41, 83 -92 (1983). [3] Gordon, R., Bender, R., Herman, G. T.: Algebraic reconstruction techniques (ART) for threedimensional electron microscopy and X-ray photography. J. theor. BioI. 29,471 -481 (1970). [4] Groetsch, C. W.: Generalized Inverses of Linear Operators. Representation and Approximation. New York: M. Dekker 1977. [5] Kaczmarz, S.: Angeniiherte Aufl6sung von Systemen linearer Gleichungen. Bull. Acad. Polon. Sci. Lett. A 35, 355 -357 (1937). [6] Marti, J. T.: On the convergence of the discrete ART algorithm for the reconstruction of digital pictures from their projections. Computing 21, 105 -Ill (1979). [7] Trummer, M. R.: Rekonstruktion von Bildern aus ihren Projektionen. Diplomarbeit, Seminar fUr Angewandte Mathematik, ETH Zurich (1979). [8] Trummer, M. R.: Reconstructing pictures from projections: on the convergence of the ART algorithm with relaxation. Computing 26, 189 -195 (1981). M. R. Trummer Seminar fUr Angewandte Mathematik Eidgenossische Technische Hochschule CH -8092 Zurich Switzerland Present address: Department of Computer Science The University of British Columbia Vancouver, B.c., Canada V6T IW5
