Wilkinson’s iterative refinement of solution with automatic step-size control for linear system of equations

In a recent paper [X.Y. Wu, R. Shao, Yiran Zhu, New iterative improvement of solution for an ill-conditioned system of linear equations based on a linear dynamic system, Comput. Math Appl. 44 (2002) 1109–1116], the authors analyzed the dynamic behavior for Wilkinson’s iterative improvement of solution and proposed a new iterative improvement of solution for an ill-conditioned system of linear equations based on a linear dynamic system. It was claimed that Wilkinson’s iterative improvement of solution can be regarded as the explicit Euler method with fixed step size h = 1 for the dynamic system. In this paper, the iterative improvement of solution with automatic step-size control is presented for linear system of algebraic equations. The convergence of the iterative procedure is shown for both stationary and non-stationary iteration, when the procedure is thought of as an iterative method. Numerical experiments are accompanied in comparison with Wilkinson’s iterative improvement.