求解对称不定线性系统的吉尔-默里强迫正定方法

程 军

(曲靖师范学院教师教育学院,云南曲靖655011)

考虑如下2×2块状线性系统

其中,A∈Rn×n是对称不定矩阵,B∈Rm×n(m≤n)满秩,即秩(B)=m,令BT表示B的转置.向量x,f∈Rn,y,g∈Rm.在此假设条件下易知线性方程组(1)的解是存在且唯一的,并且方程组的系数矩阵是非奇异的.具有形如方程组(1)的线性系统有许多实际应用背景,如计算流体力学[1-2]、电磁计算[3]、Stokes方程和二阶椭圆形的混合有限元方法,以及带约束的优化问题等[4-13].线性系统(1)中的A矩阵为对称正定或对称半正定的,有许多不同的迭代方法来求解这类问题[8-9],但是当(1,1)块矩阵A是不定矩阵的研究工作相对来说则少很多.本文针对系数矩阵(1,1)块矩阵A是不定矩阵,运用吉尔-默里强迫正定分裂方法[14]使分解成一个对称正定矩阵和一个对角矩阵,构造一个新的迭代方法,并给出该算法的收敛条件.

1 吉尔 -默里强迫正定分解算法

2 吉尔-默里强迫正定迭代方法的收敛性分析

3 数值算例

表1 吉尔-默里强迫正定迭代方法的迭代数及运行时间Table 1 Number of iterations and running time of Gill-Murry forced positive definite splitting methods

表1列出了迭代矩阵G的谱半径的值以及迭代格式(5)收敛所需要的时间.由结果可知迭代格式(5)收敛,故此算法是有效的.

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