An Optimal Self-scaling Strategy to the Modified Symmetric Rank One Updating

摘要

In the paper, the optimal self-scaling strategy to the modified symmetric rank one (HSR1) update, which satisfies the modified quasi-Newton equation, is derived to improve the condition number of the updates. The scaling factors are derived from minimizing the estimate of upper bounds on the condition number of the updating matrix. Theoretical analysis, and numerical experiments and comparisons show that introducing the optimal scaling factor into the modified symmetric rank one update preserves the positive definiteness of updates, and greatly improves the stability and numerical performance of the modified symmetric rank one algorithm.