In this dissertation, we review two types of structure-preserving doubling algorithm (SDA) and some techniques for analyzing them. We then use the techniques to study the doubling algorithm for four different nonlinear matrix equations in the critical case. We show that the convergence of the doubling algorithm is at least linear
with rate 1/2. As compared to earlier work on this topic,
the results we present here are more general, and the analysis here is much simpler. Some numerical experiments show that the SDA algorithm is feasible and effective, outperforms other iteration methods.

在本篇論文中,我們回顧了兩種型式的保結構算法, 我們只利用了一些基本的矩陣定理跟技術, 去分析這兩種保結構算法在4大類非線性矩陣方程的收斂行為, 特別在某種奇異的條件下, 我們證明了保結構算法的收斂行為是速率至少1/2的線性收斂. 跟以往的方法比較, 我們的結果更為廣泛、 限制條件更少、以及證明手法更為簡單; 透過數值實驗更可看出保結構算法比其他方法更有效率, 以及更合適的應用在這些矩陣方程。

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