Verbesserte Einzelschrittverfahren

Abstract

Zusammenfassung --AbstractVerbesserte Einzelschrittverfahren. Ausgehend von der Theorie L-regul~irer Zerlegungen erhalten wir dem Einzelschrittverfahren/ihnliche Iterationsverfahren zur L6sung linearer Gleichungssysteme A x = bmit einer M-Matrix A und vergleichen die Konvergenzgeschwindigkeiten. Im AnschluB daran werden einige der neuen Verfahren auf Matrizen mit Bandstruktur angewendet und ihre Vorztige anhand numerischer Beispiele demonstriert.Some Gauss-Seidel Like Iterative Methods. Based on the theory of L-weak-regular splittings there will be described some Gauss-Seidel like iterative methods to solve systems of linear equations A x = b with an M-Matrix A. Then we compare their rapidity of convergence, apply them especially to tridiagonal M-matrices and give some numerical examples.

New gauss–seidel like block iterative methods to solve discrete boundary value problems

“…To simplify the notation we define the stepwise sub-, super-and diagonal-matrix of The theorems proved in [3] can be transferred to the matrices A, R E L(R"') with block structure in an almost trivial way. In most cases it suffices to put the lower indices as upper ones.…”

“…mild nonlinearity or demand of a certain accuracy), in which the advantage of iterative methods will encourage us to use ABGSM [?I. An indication of better aposteriori error estimations for the new methods can be found in [3].…”

“…In the last few years mathematicians interested in solving the above mentioned problem focused their attention on direct methods [I], [5], [6], whereas [3] and this paper try to continue the former efforts to accelerate iterative methods. At the present state of the discussion the new GaussSeidel like iterative methods avowedly cannot compete for instance against the fast Poisson solver [I], [6] in this example, but it is worth while to carry on the discussion because of the special qualities of iterative methods [7] and further cognition.…”

“…In [3] we introduced a method to accelerate the Gauss-Seidel iterative method (GSM). Approximating the solution of the system of linear we restricted ourselves to quadratic coefficients matrices A with the Mproperty [8] and preferably with band structure.…”