预处理后含参数形式的SOR迭代法收敛性

The Convergence of the SOR Iterative Method with Parameters in Preconditioned

在运用SOR迭代法求解线性方程组Ax=b时,针对常见的预条件矩阵P=(I+S),本文给出预处理后迭代法的一类含参数分裂形式As=1γ{[αI-γ(L-S+L1)]-[(α-γ)I+γD1+γU]},使得分裂形式更加一般化,当α=1时就成为常见的预条件SOR迭代法。结合矩阵分析和矩阵比较定理,讨论这种含参数分裂形式下的SOR迭代法不仅能加速SOR迭代法,而且收敛速度超过常见预条件SOR迭代法,通过参数α的不同取值找到迭代法谱半径的变化趋势,得到当参数γ=α时该方法的谱半径最小,即收敛速度最快。最后给出数值例子加以验证。

In using the SOR iterative method for solving linear system Ax=b,this paper gives the SOR iterative method after preconditioned P=（I＋S） with parameters splitting form,As=1 γ{[αI-γ（L-S＋L1）]-[（α-γ）I＋γD1＋γU]}.This can make splitting form more generalized,and obtain the common preconditioned SOR iterative method when α=1.By using matrix iterative analysis and comparison theorems to discuss convergence rate of this SOR iterative method after preconditioned with parameters splitting form is not only to accelerate the SOR iterative method,but also to excel the general SOR iterative method after preconditioned.Then find the spectral radius change trend by changing parametersα,and obtain the parameter optimal selection in γ=α,at this time the convergence rate is fastest.Finally the numerical example is given to verify the conclusions