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On Optimal Parameter Not Only for the SOR Method
Applied and Computational Mathematics
Volume 8, Issue 5, October 2019, Pages: 82-87
Received: Oct. 15, 2019; Accepted: Nov. 18, 2019; Published: Nov. 22, 2019
Author
Stanislaw Marian Grzegorski, Institute of Computer Science, Lublin University of Technology, Nadbystrzycka, Lublin, Poland
Article Tools Abstract PDF (220KB)
Abstract
The Jacobi, Gauss-Seidel and SOR methods belong to the class of simple iterative methods for linear systems. Because of the parameter , the SOR method is more effective than the Gauss-Seidel method. Here, a new approach to the simple iterative methods is proposed. A new parameter q can be introduced to every simple iterative method. Then, if a matrix of a system is positive definite and the parameter q is sufficiently large, the method is convergent. The original Jacobi method is convergent only if the matrix is diagonally dominated, while the Jacobi method with the parameter q is convergent for every positive definite matrix. The optimality criterion for the choice of the parameter q is given, and thus, interesting results for the Jacobi, Richardson and Gauss-Seidel methods are obtained. The Gauss-Seidel method with the parameter q, in a sense, is equivalent to the SOR method. From the formula for the optimal value of q results the formula for optimal value of . Up to present, this formula was known only in special cases. Practical useful approximate formula for optimal value  is also given. The influence of the parameter q on the speed of convergence of the simple iterative methods is shown in a numerical example. Numerical experiments confirm: for very large scale systems the speed of convergence of the SOR method with optimal or approximate parameter  is near the same (in some cases better) as the speed of convergence of the conjugate gradients method.
Keywords
Iterative Methods for Linear Systems, Optimal Parameter for SOR Method, Positive Definite Matrices
Stanislaw Marian Grzegorski, On Optimal Parameter Not Only for the SOR Method, Applied and Computational Mathematics. Vol. 8, No. 5, 2019, pp. 82-87. doi: 10.11648/j.acm.20190805.11
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