Because the expense of estimating the optimal value of the relaxation parameter in the successive over-relaxation (SOR) method is usually prohibitive, the parameter is often adaptively controlled. In this paper, new adaptive SOR methods are presented that are applicable to a variety of symmetric positive definite linear systems and do not require additional matrix-vector products when updating the parameter. To this end, we regard the SOR method as an algorithm for minimising a certain objective function, which yields an interpretation of the relaxation parameter as the step size following a certain change of variables. This interpretation enables us to adaptively control the step size based on some line search techniques, such as the Wolfe conditions. Numerical examples demonstrate the favourable behaviour of the proposed methods.

因为在连续超松弛（SOR）方法中估计松弛参数的最佳值的费用通常是高昂的，所以通常自适应地控制该参数。在本文中，提出了新的自适应SOR方法，该方法适用于各种对称正定线性系统，并且在更新参数时不需要其他矩阵矢量积。为此，我们将SOR方法视为一种用于最小化某个目标函数的算法，该算法可将松弛参数解释为变量变化后的步长。这种解释使我们能够基于某些线搜索技术（例如Wolfe条件）来自适应地控制步长。数值算例表明了所提出方法的良好行为。