Riccati型矩阵方程的迭代算法和加倍算法：比较介绍
Iterative and doubling algorithms for Riccati-type matrix equations: a comparative introduction
摘要：
本文综述了一类求解李雅普诺夫型和黎卡提型方程的算法，它们都是通过\emph{doubling}的思想联系在一起的：它们通过一种重复平方构造了另一个自然产生的不动点迭代$(X_h)$的解$Q_k = X_{2^k}$.我们考虑的方程是Stein方程$X - A^*XA=Q$、Lyapunov方程$A^*X+XA+Q=0$、离散时间代数Riccati方程$X=Q+A^*X(I+GX)^{-1}A$、连续时间代数Riccati方程$Q+A^*X+XA-XGX=0$、回文二次矩阵方程$A+QY+A^*Y^2=0$和非线性矩阵方程$X+A^*X^{-1}A=Q$。我们绘制这些算法之间的比较，突出它们之间的连接和其他算法，如子空间迭代，并讨论其理论中的开放问题。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
