A note on Wilkinson's iterative refinement of solution with automatic step-size control

Abstract

In a more recent paper [X. Wu and Y.L. Fang, Wilkinson's iterative refinement of solution with automatic step-size control for linear system of equations, Appl. Math. Comput. 193 (2007), pp. 506–513], the authors proposed an iterative improvement of solution with automatic step-size control for a linear system of algebraic equations. The convergence analysis of the iterative procedure is shown for both stationary and non-stationary iterative formulas, based on the accurate coefficient matrix A and its inverse. However, in each iteration, A is approximated by the LU decomposition in floating-point arithmetic, and then its inverse is also approximated by a matrix B, although the role of B is realized by solving something like Az=r from the LU factorization of A. This point should be addressed that usually BA is not equal to an identity matrix I, in particular, when A is ill-conditioned. Therefore, in this paper, a supplementary convergence analysis is presented based on the approximation.

Keywords:

• Wilkinson's iterative refinement of solution
• ill-conditioned linear system of algebraic equations
• automatic step-size control
• embedded pairs of Wilkinson's iterative improvement
• convergence analysis

2000 AMS Subject Classifications :

• Primary 65F22
• 65F10

Acknowledgements

The authors are grateful to the editorial board and referees for their valuable comments which led to the improvement of this paper. The research is supported by the Natural Science Foundation of China under Grant 10771099.