Über einige Methoden der Relaxationsrechnung

Summary

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.