一类混合型Lyapunov矩阵方程的对称正定解

本文研究了一类混合型Lyapunov矩阵方程的对称正定解问题。首先将此方程转化为等价的含参矩阵方程,然后运用矩阵分解和紧凸集上不动点定理,给出了方程具有对称正定解的一些必要和充分条件:其次建立两种求方程对称正定解的参数迭代算法,分析了迭代的收敛性及参数的选取方法,并指出这两种算法的适应性和特点;数值算例表明上述算法的可行性和有效性,并对比出两种迭代的敛速。

The Symmetric Positive Definite Solution to a Class of Mixed-type Lyapunov Matrix Equations

In this paper,we study the problem about the symmetric positive definite solution to a class of mixed-type Lyapunov matrix equations.By firstly transforming this equation into a matrix equation with parameter equivalently,and then applying decomposition of a matrix and a fixed point theorem on compact convex set,some necessary and sufficient conditions for the existence of a symmet- ric positive definite solution of this equation are derived.Next,we construct two iterative algorithms with parameter to find a symmetric positive solution of the matrix equation,the convergence of the algorithms and parameter choosing method are analyzed,and we point out the adaptability and char- acteristics of the two algorithms.Finally,a numerical example shows that above algorithms are feasible and efficient.