Alternating-directional Doubling Algorithm for M-Matrix Algebraic Riccati Equations
Alternating-directional Doubling Algorithm for M-Matrix Algebraic Riccati Equations
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DOI：
10.1137/110835463
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
Wei-guo Wang;Wei-chao Wang;Ren-Cang Li
中科院分区：
文献类型：
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作者：
Wei-guo Wang;Wei-chao Wang;Ren-Cang Li
提出了一种计算$M$矩阵代数Riccati方程（MARE）唯一最小非负解的新加倍算法——交替方向加倍算法（ADDA）。理论分析和数值实验表明，ADDA总是比现有的两种加倍算法：Guo， Lin, and Xu （number）的SDA更快。数学。， 103 (2006), pp. 393-412)和SDA-ss的Bini， Meini, and Poloni(数字。数学。， 116(2010)，第553-578页)为同样的目的。还证明了所有三种方法都能够提供最小的非负解，并且具有由MARE的定义系数矩阵保证的入口相对精度。这三种加倍算法只在初始设置上有所不同，它们对应于一般双线性（也称为Mobius）变换的三种特殊情况。说明在所有双线性变换产生的所有可能的加倍算法中，ADDA是最好的。
A new doubling algorithm—the alternating-directional doubling algorithm (ADDA)—is developed for computing the unique minimal nonnegative solution of an $M$-matrix algebraic Riccati equation (MARE). It is argued by both theoretical analysis and numerical experiments that ADDA is always faster than two existing doubling algorithms: SDA of Guo, Lin, and Xu (Numer. Math., 103 (2006), pp. 393-412) and SDA-ss of Bini, Meini, and Poloni (Numer. Math., 116 (2010), pp. 553-578) for the same purpose. Also demonstrated is that all three methods are capable of delivering minimal nonnegative solutions with entrywise relative accuracies as warranted by the defining coefficient matrices of a MARE. The three doubling algorithms, differing only in their initial setups, correspond to three special cases of the general bilinear (also called Mobius) transformation. It is explained that ADDA is the best among all possible doubling algorithms resulted from all bilinear transformations.