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

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.