The G-transformation is an efficient method for solving the weighted least squares problems. However, the underflows were not considered in the original G-transformation. In order to keep its stability, some authors proposed three specified scaling strategies for guarding against the underflows. Note that these three specific strategies are not easy to be implemented in actual operations, in this paper we present a self-scaling G-transformation (SSGT) which not only avoids these specified scaling strategies, but maintain the stability of operations. Complexity analysis of our self-scaling G-transformation shows that its cost of computation is less than that of the G-transformation, which implies the high efficiency of our proposed SSGT. The stability of the SSGT was theoretically confirmed by a detailed error analysis. Some numerical experiments are performed to illustrate the effects of the self-scaling strategy and show the attractiveness of the SSGT method when solving sparse weighted least squares problems.

G变换是解决加权最小二乘问题的有效方法。但是，在原始G转换中未考虑下溢。为了保持其稳定性，一些作者提出了三种指定的缩放策略以防止下溢。请注意，这三种特定策略在实际操作中不容易实现，在本文中，我们提出了一种自缩放G变换（SSGT），它不仅避免了这些指定的缩放策略，而且还保持了操作的稳定性。对我们自定标的G变换的复杂性分析表明，它的计算成本低于G变换的计算成本，这表明我们提出的SSGT具有很高的效率。理论上，通过详细的误差分析可以确认SSGT的稳定性。