求解矩阵方程AX=B的多步迭代算法

Multi-step iterative algorithm for solving matrix equation AX=B

为求解矩阵方程AX=B的一般解及其最小二乘问题,提出了一种多步迭代算法,给出并证明了由该算法产生的矩阵序列收敛于矩阵方程AX=B及其最小二乘问题的一般解和最小Frobenius范数解的条件。通过理论分析和数值实验证明了该算法的收敛性和有效性;数值结果表明,该算法的收敛速度比基于梯度的迭代算法更快。

In order to solve the general solution of matrix equation AX=B and its least squares problem,a multi-step iterative algorithm is proposed.The conditions that the matrix sequence generated by the multi-step iterative algorithm converges to the general solution of the matrix equation AX=B and the minimum Frobenius norm solution of its least squares problem are given and proved;The convergence and effectiveness of the algorithm are proved by theoretical analysis and numerical experiments.Numerical results show that the multi-step iterative algorithm has a faster convergence speed than the gradient-based iterative algorithm proposed.