廣義分裂下的預(yù)處理Gauss-Seidel迭代法收斂性的討論

周 婷，張仕光

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廣義分裂下的預(yù)處理Gauss-Seidel迭代法收斂性的討論

*周 婷，張仕光

(衡水學(xué)院數(shù)學(xué)與計(jì)算機(jī)學(xué)院，河北，衡水 053000)

運(yùn)用Gauss-Seidel迭代法解線(xiàn)性方程組，討論了在一類(lèi)預(yù)條件矩陣下的Gauss-Seidel迭代法的收斂性。在更廣義的分裂條件下，對(duì)預(yù)條件Gauss-Seidel迭代法和相應(yīng)的Gauss-Seidel迭代法的收斂性進(jìn)行了比較，得到了比較定理。最后給出數(shù)值例子驗(yàn)證了所得到的主要結(jié)論。

預(yù)條件；-矩陣；-矩陣；Gauss-Seidel迭代法

考慮線(xiàn)性方程組

， (1)

1 預(yù)備知識(shí)

引理 1[7]設(shè)是一個(gè)-矩陣，那么下面幾個(gè)命題等價(jià)：

(1)是非奇異-矩陣；

(3)的所有主子式都是正的.

2 主要結(jié)果

則預(yù)條件Gauss-Seidel迭代法的迭代矩陣為

.

因?yàn)槭且粋€(gè)非奇異-矩陣，

3 數(shù)值例子

考慮滿(mǎn)足定理1條件的方程組(1)的系數(shù)矩陣，

令

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The convergence disscussion of the preconditioned Gauss-Seidel iterative method with a more general splitting

*ZHOU Ting, ZHANG Shi-guang

(College of Mathematics and Computer Science Hengshui University, Hengshui, Hebei 053000, China)

Using the Gauss-Seidel iterative method for the solution of the linear equations, the convergence of the Gauss-Seidel iterative method is discussed under a type of preconditioned matrix. With a more general splitting, we compare the convergence of the preconditioned Gauss-Seidel iterative method and the corresponding Gauss-Seidel iterative method. Furthermore, we get some comparison theorems. Finally, a numerical example is given to illustrate the validity of the conclusions.

precondition;-matrix;-matrix; Gauss-Seidel iterative method

1674-8085(2012)03-0013-03

O241.6

A

10.3969/j.issn.1674-8085.2012.03.003

2012-03-06；

2012-04-11

河北省高等學(xué)校科學(xué)研究計(jì)劃項(xiàng)目(Z2010188);衡水學(xué)院2011年科學(xué)研究項(xiàng)目(2011026)

*周 婷(1976-)，女，山東臨朐人，碩士，主要從事數(shù)值計(jì)算方法及其應(yīng)用研究(E-mail: zhouting7606@163.com);

張仕光(1975-)，男，山東平度人，講師，碩士，主要從事廣義逆理論及應(yīng)用研究(E-mail: shiguang08@yahoo.com.cn).