基于交替投影算法求解单变量线性约束矩阵方程问题

ALTERNATING PROJECTION ALGORITHM FOR SINGLE VARIABLE LINEAR CONSTRAINTS MATRIX EQUATION PROBLEMS

研究如下线性约束矩阵方程求解问题:给定A∈R^(m×n),B∈R^(n×p)和C∈R^(m×p),求矩阵X∈R(?)R^(n×n)"使得A×B=C以及相应的最佳逼近问题,其中集合R为如对称阵,Toeplitz阵等构成的线性子空间,或者对称半(ε)正定阵,(对称)非负阵等构成的闭凸集.给出了在相容条件下求解该问题的交替投影算法及算法收敛性分析.通过大量数值算例说明该算法的可行性和高效性,以及该算法较传统的矩阵形式的Krylov子空间方法(可行前提下)在迭代效率上的明显优势,本文也通过寻求加速技巧进一步提高算法的收敛速度.

We consider the following linear constrained matrix equation problem： given matrices A∈Rm×n,B∈Rn×p,and C∈m×p, find matrix X∈RCRn×n such that AXB = C, and the associate optimal approximation problem. The matrix set 7~ are considered as lin- ear subspaces which are composed of symmetric matrices, Toeplitz type matrices and so on, or closed cone sets which are composed of symmetric positive semi（c）-definite matri- ces, （symmetric） nonnegative matrices and so on. A alternating projection algorithm and some convergence acceleration techniques are presented to solve the proposed problem in the premise of consistent and some convergence results of the algorithm are proved. Numerical experiments are performed to illustrate the applicability of the algorithm and a comparison with some existing Krylov subspace methods （in the premise of consistent） is also given.