多变量LME一种异类约束最小二乘解的迭代算法

摘要

基于求多矩阵变量线性矩阵方程(LME)异类约束解的迭代算法,通过构造等价的线性矩阵方程组,建立了求多变量LME的一种异类约束最小二乘解的迭代算法.该算法不要求等价线性方程组的系数矩阵正定、可逆或者列满秩,因此该算法总是可行的.不考虑舍入误差时,该算法可在有限步计算后求得多变量LME的一组异类约束最小二乘解；选取特殊的初始矩阵时,可求得多变量LME的极小范数异类约束最小二乘解.此外,还可在多变量LME的异类约束最小二乘解集合中给出指定矩阵的最佳逼近矩阵.算例表明,迭代算法是有效的.%Based on the modified conjugate gradient method to find the different constrained solution of a multi-variables linear matrix equation, an iterative method is constructed to find the different constrained least square solution by constructing the equivalent linear matrix equations. Positive definiteness, reversibility and full column rank of the coefficient matrix of the equivalent linear matrix are not required for the iteration algorithm. So the iterative method is always feasibility. By this iterative method, the different constrained least square solutions can be obtained within finite iterative steps in the absence of round off errors, and the different constrained least square solution with least-norm can be got by choosing special initial matrix. In addition, the optimal approximation matrix to any given matrix can be obtained in the set of the different constrained least square solutions. The numerical examples show that the iterative method is efficient.