Neumann Series-Based Adaptation of Stationary Iterative Methods for Solution of Linear Algebraic Systems of Equations

Abstract

Solution of linear algebraic systems of equations plays a major role in real life problems. The iteration matrices of both the Jacobi and Gauss-Seidel schemes are split into strictly lower triangular, diagonal and strictly upper triangular matrices. The resulting Jacobi and Gauss-Seidel schemes are then expressed in terms of Neumann series. Taking the initial approximation to be the zero-column vector of the same dimension as the right-hand side constant column vector, a new series is obtained in terms of each of the iteration matrices. A recursive scheme is developed through the application of Neumann series. This series was then approximated to give the new scheme which solves linear systems with various coefficient matrices. The new iterative schemes obtained were verified with examples