求解對稱不定線性系統的吉爾-默里強迫正定方法

程 軍

(曲靖師范學院教師教育學院,云南曲靖655011)

考慮如下2×2塊狀線性系統

其中,A∈Rn×n是對稱不定矩陣,B∈Rm×n(m≤n)滿秩,即秩(B)=m,令BT表示B的轉置.向量x,f∈Rn,y,g∈Rm.在此假設條件下易知線性方程組(1)的解是存在且唯一的,并且方程組的系數矩陣是非奇異的.具有形如方程組(1)的線性系統有許多實際應用背景,如計算流體力學[1-2]、電磁計算[3]、Stokes方程和二階橢圓形的混合有限元方法,以及帶約束的優化問題等[4-13].線性系統(1)中的A矩陣為對稱正定或對稱半正定的,有許多不同的迭代方法來求解這類問題[8-9],但是當(1,1)塊矩陣A是不定矩陣的研究工作相對來說則少很多.本文針對系數矩陣(1,1)塊矩陣A是不定矩陣,運用吉爾-默里強迫正定分裂方法[14]使分解成一個對稱正定矩陣和一個對角矩陣,構造一個新的迭代方法,并給出該算法的收斂條件.

1 吉爾 -默里強迫正定分解算法

2 吉爾-默里強迫正定迭代方法的收斂性分析

3 數值算例

表1 吉爾-默里強迫正定迭代方法的迭代數及運行時間Table 1 Number of iterations and running time of Gill-Murry forced positive definite splitting methods

表1列出了迭代矩陣G的譜半徑的值以及迭代格式(5)收斂所需要的時間.由結果可知迭代格式(5)收斂,故此算法是有效的.

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