Chebyshev多项式加速不同分块的SOR迭代方法 Alternative Title Chebyshev polynomial acceleration of different SOR iterative method 段利英 Thesis Advisor 郑兵 2008-05-24 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword Chebyshev多项式 2-block SOR迭代方法 3-block SOR迭代方法 最优外推迭代法 4-block SOR迭代方法 Abstract 许多科学计算问题最终将转化为线性系统的求解问题. 而线性最小二乘问题[26]是计算数学一个重要的领域, 也是一个非常活跃的研究领域. 它在大地测量、摄影测量、结构分析、分子结构学等的科学计算中均有广泛的应用. 　　继古典的迭代法如:Jacobi方法、GS方法、SOR方法、SSOR方法后, 人们又提出了它们的分块迭代法如: BJ方法、BGS方法、BSOR方法、SBSOR方法及AOR方法([10], [39], [41])GSOR方法[42]、SOR-like方法([23],[24],[27])、AG-SOR方法[15]等. 分裂迭代法来解决线性最小二乘问题是非常有效的. 但对于病态的方程组, 一些迭代法就有些不足. 在迭代法中, 迭代格式收敛性及收敛速度就成为关键问题. 下面我们就对一些分裂迭代法进行加速, 通过讨论加速后迭代矩阵的收敛性来证明我们的方法优于加速前的迭代法, 并对病态的迭代矩阵同样有效. 　　本文用Chebyshev多项式加速不同分块的SOR迭代法, 给出了三种新的迭代格式来解决线性最小二乘问题. 通过讨论它们的收敛性, 并与加速前的迭代法以及最优外推法的收敛性做比较, 我们发现加速后的方法优于加速前的方法, 并且收敛速度是相应最优外推法的二倍. 最后, 我们比较了两种新迭代格式, 并给出了数值例子. Other Abstract Many scientific computing problems eventually turn into a linear system to solve　the problems. And linear least squares problems [26] is a important areas in mathematical calculation, also is a very active research field. It is　applied to many scientific　computing areas such as geodesy, photogrammetry, structural analysis, and structural molecular. 　Following the classic method such as: Jacobi, GS, SOR, SSOR methods, people　put forward their block iterative methods such as: BJ method, BGS method, BSOR　method, SBSOR method, AOR ([10], [39], [41])method, GSOR method [42],　SORlike　approach ([23], [24], [27]), AGSOR method [15], etc. Splitting iterative method is　very effective to solve linear least squares problems, but for morbid equation, a　number　of iterative methods have their deficient. In the iterative method, the convergence and　convergence rate　play a major role. Below we shall accelerate some different iterative　methods. Through the discussions of the　convergence of accelerated iterative matrix,　we prove that our method is superior to the iterative method before, and　is efficient for　sick equations. 　 　In this paper, we apply the Chebyshev polynomials to the different SOR iterations,　and present three new iterative　format to solve linear least squares problems. Through　discussing of the convergence, and comparing with the　iteration method before and the　extrapolation method, we found that the new method is superior to the method before. And the corresponding optimal convergence rate is twice the extrapolation method. Finally, we compare the two new methods and give numerical examples. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/225730 Collection 数学与统计学院 Recommended CitationGB/T 7714 段利英. Chebyshev多项式加速不同分块的SOR迭代方法[D]. 兰州. 兰州大学,2008.
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