Alternating-directional Doubling Algorithm for M-Matrix Algebraic Riccati Equations

Abstract

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.