Extended version with the analysis of dynamic system for iterative refinement of solution

Abstract

The extended version with the analysis of dynamic system for Wilkinson's iteration improvement of solution is presented in this paper. It turns out that the iteration improvement can be viewed as applying explicit Euler method with step size h=1 to a dynamic system which has a unique globally asymptotically stable equilibrium point, that is, the solution x*=A ⁻¹ b of linear system Ax=b with non-singular matrix A. As a result, an extended iterative improvement process for solving ill-conditioned linear system of algebraic equations with non-singular coefficients matrix is proposed by following the solution curve of a linear system of ordinary differential equations. We prove the unconditional convergence and derive the roundoff results for the extended iterative refinement process. Several numerical experiments are given to show the effectiveness and competition of the extended iteration refinement in comparison with Wilkinson's.

Keywords:

• extended iterative improvement of solution
• ill-conditioned linear system of algebraic equations
• dynamic system
• iteration method
• roundoff analysis
• Wilkinson's refinement of solution

2000 AMS Subject Classification :

• Primary: 65F22
• 65F10

Acknowledgements

The authors sincerely thank the editors and referees for their kind and valuable comments of revision which improved the presentation of the paper and made the contents of the paper more precise. This project was supported by the Natural Science Foundation of China under grants 60573157 and 10771099.