A New Iterative Method for Solving Linear Systems of Equations

Abstract

The Jacobi and Gauss-Seidel iterative methods are among iterative methods for solving linear system of equations. In this paper, a new iterative method is introduced, it is based on the linear combination of old and most recent calculated solutions. The new method can be considered as a general method, where the Jacobi and Gauss-Seidel methods as two special cases of it. Some convergence properties are studied, and numerical examples are given to show the effectiveness of the new method. When Jacobi method converges, the new method can be used to accelerate the convergence. In special cases, when one of the two iterative methods, Jacobi or Gauss-Seidel, diverges, the new method can be used to obtain convergence.

Key words and phrases. Jacobi method; Gauss-Seidel method; Linear Systems; Iterative Methods; Convergence.

2000 Mathematics Subject Classification. 65F10, 93C05