On Optimal Parameter Not Only for the SOR Method

Abstract

The Jacobi, Gauss-Seidel and SOR methods belong to the class of simple iterative methods for linear systems. Because of the parameter ω, 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 ω. Up to present, this formula was known only in special cases. Practical useful approximate formula for optimal value ω 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 ω is near the same (in some cases better) as the speed of convergence of the conjugate gradients method.

New iterative methods for nonlinear operators as concerns convex programming applicable in differential problems, image deblurring, and signal recovering problems

“…For stability of WJ and SOR method in solving linear system () generated from the discretization of the consideration problem (), the step sizes of time play an important role of the stability needed. The discussion on the stability of WJ and SOR in solving linear system () can be found in Yambangwai et al 31 and Grzegorski 32 …”

“…To find the solution of linear system (), we manipulate this linear system into the form of fixed point equation $$$ \mathbf{u}=\mathcal{T}\left(\mathbf{u}\right) $$$ (see on Table 2). For example, the well‐known weight Jacobi (WJ), successive over relaxation (SOR), and Gauss‐Seidel (GS, SOR with $$$ \omega =1 $$$) methods 30–32 present the linear system () into the form of fixed point equation as $$$ {\mathcal{T}}^{\mathrm{WJ}}\left(\mathbf{u}\right)=\mathbf{u},{\mathcal{T}}^{\mathrm{SOR}}\left(\mathbf{u}\right)=\mathbf{u} $$$ and $$$ {\mathcal{T}}^{\mathrm{GS}}\left(\mathbf{u}\right)=\mathbf{u} $$$, respectively.…”

“…The different way of rearranging linear systems (32) into the form u = T(u). From Equation ( 31), matrix A is square and symmetric positive definite.…”

“…Traditionally, iterative methods have been presented in solving the solution of linear systems (31). The well-known weight Jacobi (WJ) and successive over relaxation (SOR) methods 31,32 are chosen to exemplify here (see on Table 1).…”

“…The discussion on the stability of WJ and SOR in solving linear system (31) can be found in Yambangwai et al 31 and Grzegorski. 32 Let consider the linear system…”