用TOR方法求解最小二乘问题收敛域

Convergence Area of TOR Iterative Method for Solving Least Squares Problems

为了求解大型稀疏超定线性方程组 ,通常人们都是求它的极小范数最小二乘解 很多直接和间接方法被人们研究 在这些方法中求解最小二乘问题的通常的SOR ,SSOR ,TOR等迭代方法发挥了重要作用 ,被一些作者建议并研究 ,笔者讨论了用TOR方法求解最小二乘问题的收敛域 ,首先导出了块JACOBI迭代矩阵的特征值集合与TOR迭代矩阵的特征值集合之间的关系 接着用比较直接的方法得到用TOR方法求解最小二乘问题收敛域和发散域 ,结果有所改善 最后给出了算例 比较了对于ω、γ不同选取 ,TOR方法的收敛速度 选取适当的参数值时 ,可使TOR迭代法的收敛速度加快 ,且在同一谱半径下 ,当ω <γ时的收敛速度比ω >

In order to solve large size sparse overdetermined linear systems, the common practice is to find out its least squares solution with minimal norm. Many direct and indirect methods are discussed, among which SOR, SSOR, TOR iterative methods for solving Least Squares Problems play a very important role. The paper discusses the convergence area of TOR iterative method for solving least squares problems. We first derive relationships between the sets of eigenvalues of the block Jacobi iteration matrix and that of TOR iteration matrix. Then convergence and divergence area of TOR iterative method for solving least squares problems are obtained by more straightforward methods. The results are improved compared with those in the first paper cited. At last, a numerical example is given. And the paper also compares the convergence rates of TOR method for various ω、γ . When parameters are selected properly, the convergence rate of TOR iterative method can be improved and in the same spectral radii, the convergence rate when ω<γ is faster than that when ω>γ .