论文标题
Riccati-Type矩阵方程的迭代和加倍算法：比较介绍
Iterative and doubling algorithms for Riccati-type matrix equations: a comparative introduction
论文作者
论文摘要
我们回顾了lyapunov-和riccati-type方程式的算法家庭，它们通过\ emph {floupping}的想法彼此相关。 我们考虑的方程是stein方程$ x-a^*xa = q $，lyapunov方程$ a^*x+xa+q = 0 $，离散时间algebraic riccati方程$ x = q+a^*x（i+gx） palindromic二次矩阵方程$ a+qy+a^*y^2 = 0 $，而非线性矩阵方程$ x+a^*x^{ - 1} a = q $。我们在这些算法中进行了比较，突出了它们与其他算法（例如子空间迭代）之间的联系，并讨论其理论中的开放问题。
We review a family of algorithms for Lyapunov- and Riccati-type equations which are all related to each other by the idea of \emph{doubling}: they construct the iterate $Q_k = X_{2^k}$ of another naturally-arising fixed-point iteration $(X_h)$ via a sort of repeated squaring. The equations we consider are Stein equations $X - A^*XA=Q$, Lyapunov equations $A^*X+XA+Q=0$, discrete-time algebraic Riccati equations $X=Q+A^*X(I+GX)^{-1}A$, continuous-time algebraic Riccati equations $Q+A^*X+XA-XGX=0$, palindromic quadratic matrix equations $A+QY+A^*Y^2=0$, and nonlinear matrix equations $X+A^*X^{-1}A=Q$. We draw comparisons among these algorithms, highlight the connections between them and to other algorithms such as subspace iteration, and discuss open issues in their theory.
