## 线性方程组求解的迭代法及Matlab的应用

Iteration Method for Solving Linear Systems and the Application of Matlab
Abstract: The iteration method is one of the main methods to solve linear equations. The thought of iterating, firstly, is changing the linear equations into an iterative formula. In addition, using the arbitrary given initial values and the iterative formula to iterate. Then we get the solution of the equations by the way of approaching gradually. This paper introduced some elementary iteration methods of stationary iteration and nonstationary iteration firstly, like Jacobi iteration, Gauss-Seidel iteration, SOR iteration, steepest descent method, and Conjugate gradient method. Some convergence theorems of stationary iteration are given. Finally, it shown kinds of Matlab process and used them to solving practical questions.
Key words: Linear equations; Jacobi iteration; Gauss-Seidel iteration; SOR iteration; Steepest descent method; Conjugate gradient method

1    引言    1
1.1    课题的目的和意义    1
1.2    国内外研究现状与发展趋势    1

2    定常迭代法    3
2.1    雅可比迭代法    4
2.2    高斯－赛德尔迭代法    4
2.3    超松弛迭代法    5
2.4    迭代的收敛性分析    6
2.5    实例    7
3    不定常迭代    9
3.1    最速下降法    9
3.2    共轭梯度法    10
3.3    实例    11
4    Matlab在定常迭代与不定常迭代中的应用    12
4.1    雅可比迭代法的程序    12
4.2    高斯-赛德尔迭代的程序    13
4.3    超松弛迭代的程序    13
4.4    最速下降法的程序    14
4.5    共轭梯度法的程序    14
4.6    Matlab实现的实例    15
4.6.1 定常迭代的收敛速度的比较    15
4.6.2 超松弛迭代法松弛因子的选择    16
4.6.3 不定常迭代的收敛速度的比较    18

1    引言
1.1    课题的目的和意义

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