In this paper, a new iterative refinement for ill-conditioned linear systems is derived based on discrete gradient methods for gradient systems. It is proved that the new method is convergent for any initial values irrespective of the choice of the stepsize h. Some properties of the new iterative refinement are presented. It is shown that the condition number of the coefficient matrix in the linear system to be solved in every step can be improved significantly compared with Wilkinson’s iterative refinement. The numerical experiments illustrate that the new method is more effective and efficient than Wilkinson’s iterative refinement when dealing with ill-conditioned linear systems.

中文翻译：

---

基于离散梯度的病态线性系统的迭代优化

本文基于梯度系统的离散梯度方法，推导了病态线性系统的一种新的迭代细化方法。事实证明，无论步长h的选择如何，新方法都可以收敛于任何初始值。提出了新的迭代细化的一些属性。结果表明，与威尔金森的迭代细化相比，线性系统中每一步要求解的系数矩阵的条件数都可以得到显着改善。数值实验表明，在处理病态线性系统时，该新方法比Wilkinson的迭代改进更为有效。