Über einige Methoden der Relaxationsrechnung

  • Eduard Stiefel


The general outlines of the so-called relaxation technique are developed. By “relaxation” is meant every “step-by-step procedure” for solving systems of linear equations based on minimizing quadratic forms. After a short discussion of the trial methods developed bySouthwell and his school, allowing full leeway to the intuition of the computing person, the general mathematical background is treated. § 3, 4 are the central parts of the paper. After a study of the gradient method it is shown that relaxation methods are not necessarily successive approximations taking an infinite number of steps but that it is possible to speed up convergence such that the desired result is reached in a finite number of steps. These methods may be suitable for use on sequence-controlled computing machines. Special consideration is given to the well-known fact that relaxation very quickly smoothes the trial function but that it may be a combersome task to get rid of the remaining smooth residual distribution.


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  1. 1).
    R. V. Southwell,Relaxation Methods in Engineering Science Clarendon, Oxford 1946);Relaxation Methods in Theoretical Physics (Clarendon, Oxford 1946).Google Scholar
  2. 2).
    H. Cross,Analysis of Continuous Frames by Distributing Fixed-End Moments, Proc. Amer. Soc. chem. Eng.1930);Numerical Methods of Analysis in Engineering (Symposium Illinois Institute of Technology, Macmillan, New York 1949).Google Scholar
  3. 3).
    Man vergleiche auch:L. Collatz,Numerische Behandlung von Differentialgleichungen (Springer, Berlin 1951), S. 106 und 295. Ferner:G. Temple,The general Theory of Relaxation Methods Applied to Linear Systems, Proc. Roy. Soc. London [A],169, 476–500 (1939). Dort sind das allgemeine Verfahren der Relaxationsrechnung (Abschnitt 2) und das Verfahren des stärksten Abstieges (Abschnitt 4) geschildert und Konvergenzbeweise gegeben.Google Scholar
  4. 4).
    Allgemeine Konvergenz- und Fehlertheorie findet man bei:J. von Neumann undH. H. Goldstine,Numerical Inverting of Matrices of high Order, Bull. Amer. Math. Soc.53, Nr. 11 (1947).Google Scholar

Copyright information

© Birkhäuser-Verlag 1952

Authors and Affiliations

  • Eduard Stiefel
    • 1
  1. 1.ETH.Zürich

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