秩亏最小二乘问题的预条件AOR迭代法

Study of 2-block AOR iterative method for rank deficient least squares problems

秩亏最小二乘问题来源于统计学问题、最优化问题等科学与工程计算领域。由于实际问题所对应的线性方程组的系数矩阵的阶数比较大,且秩亏,换句话说,矩阵A是不可逆的,使其求解变得更为复杂,因此,研究求解秩亏最小二乘问题的高效方法就变得尤为重要。为了求解秩亏最小二乘问题,在预处理基础上提出了二分块的AOR迭代法;研究了新建立的AOR迭代法的收敛性和最优参数的选取,得到了一些相关的定理。数值例子验证了所给方法的可行性。数值实验和理论都表明:新的AOR方法的计算格式更加简单、收敛速度快、并具有广泛的适用性,同时行满秩矩阵A1的选取要比文献[8]中可逆方阵A11的选取更方便。

Rank-deficient least squares problems arise from many scientific and engineering computations such as statistics, optimal problem and so on. In the practical problems, since the order number of corresponding coefficient matrix of linear equations is larger, and the rank of matrix is a deficit. In other words, matrix A is irreversible. Then solving process is become more complex. So it is very important to study of the suitable iterative methods for rank-deficient least squares problems. For solving the least square problems with rank-deficient, the 2-block AOR method by preconditioning technique was given. The convergence analysis of the new AOR method and the choice of optimal relaxation parameters were studied. The corresponding theorems were gotten. Numerical examples showed the effectiveness of new method. It suggests that the new iterative AOR method is simpler, faster in convergence speed, more extensive applicability than the method in [83. Meanwhile, matrix A1 is full row rank, it is more convenient than the requirement of All in [18].