Numerical stability of iterative refinement with a relaxation for linear systems

TL;DR

This paper analyzes the numerical stability of Wilkinson's iterative refinement method with relaxation for solving linear systems, confirming that the standard choice omega=1 offers optimal stability through theoretical and MATLAB-based numerical tests.

Contribution

It extends stability analysis to IR(omega) with relaxation, providing theoretical insights and numerical validation for the optimality of omega=1.

Findings

Omega=1 yields the best numerical stability.

Iterative refinement with relaxation converges under certain conditions.

Numerical tests confirm theoretical stability results.

Abstract

Stability analysis of Wilkinson's iterative refinement with a relaxation IR(omega) for solving linear systems is given. It extends existing results for omega=1, i.e., for Wilkinson's iterative refinement. We assume that all computations are performed in fixed (working) precision arithmetic. Numerical tests were done in MATLAB to illustrate our theoretical results. A particular emphasis is given on convergence of iterative refinement with a relaxation. Our tests confirm that the choice omega=1 is the best choice from the point of numerical stability.