线性矩阵方程异类约束最小二乘解的迭代算法

摘要

The constrained solution to multi-variable linear matrix equations has been widely used in parametre identification,structural design,vibration theory,automatic control theory and so on. Based on the iterative method for the same constrained least square solution of the linear matrix equation, an iterative method is constructed for different constrained least square solution of the linear matrix equation by constructing equivalent linear matrix equations. And the convergence of this method can be proved. By this method, a different constrained least square solution can be obtained within finite iterative steps in the absence of roundoff errors, and the different constrained least square solution with least-norm can be obtained by choosing special initial matrices. In addition, the optimal approximation matrix to any given matrix can be obtained in the set of different constrained least square solutions. Examples show that the method is efficient.%多矩阵变量线性矩阵方程(LME)约束解的计算问题在参数识别、结构设计、振动理论、自动控制理论等领域都有广泛应用.本文借鉴求线性矩阵方程(LME)同类约束最小二乘解的迭代算法,通过构造等价的线性矩阵方程组,建立了求多矩阵变量LME的一种异类约束最小二乘解的迭代算法,并证明了该算法的收敛性.在不考虑舍入误差的情况下,利用该算法不仅可在有限步计算后得到LME的一组异类约束最小二乘解,而且选取特殊初始矩阵时,可求得LME的极小范数异类约束最小二乘解.另外,还可求得指定矩阵在该LME的异类约束最小二乘解集合中的最佳逼近解.算例表明,该算法是有效的.