一类变异型Chebyshev-Halley迭代法的收敛性
发布时间：2018-10-30 20:46
【摘要】：现代的社会是信息化高速发展的社会,求解一些迭代问题同样更要与信息化同步.因此,怎样提高迭代速度、增加迭代范围、减少计算工作量,都是计算数学中至关重要的.本文通过改变原有的迭代收敛判据、扩大原有的迭代收敛范围,快速地达到收敛点,并对半局部收敛的迭代方法进行探索.具体内容如下：第一章主要介绍了Chebyshev迭代法和Halley迭代法的发展历史以及与Chebyshev-Halley型迭代法相关的预备知识,包括基础概念、收敛阶、收敛判据及Banach空间的相关结论。给出本文的主要思想及主体论文的解法过程.最后给出了本文的结构框架.第二章研究了一类变异型Chebyshev-Halley迭代法的收敛性.给出了在满足条件时的迭代法收敛性判据及半局部收敛性的证明,最后分析了参数α的变化对收敛半径的影响,以提供某种参数选择的依据.首先把Chebyshev-Halley型迭代式分解成两部分,第一部分为简单的Newton迭代,可以直接根据文献知识解决；第二部分是难点部分,主要利用一组单调数列简化迭代步骤并给出相应的结果.第三章研究了在中心Lipschitz条件|下,用循环数列研究Chebyshev-Halley型迭代法的收敛性及迭代误差.然后,进一步探索在一阶可导且二阶不可导的条件下,将类N秫on法运用到Chebyshev-Halley型迭代中,在理论上可以减少迭代计算量并简化迭代条件.
[Abstract]:Modern society is a society with rapid development of information, so solving some iterative problems also needs to be synchronized with informatization. Therefore, how to increase the speed of iteration, increase the scope of iteration, and reduce the computational workload are of great importance in computational mathematics. In this paper, by changing the original criterion of iterative convergence, we extend the original range of iterative convergence, reach the convergence point quickly, and explore the semi-locally convergent iterative method. The main contents are as follows: in the first chapter, the history of Chebyshev iterative method and Halley iterative method and the preparatory knowledge related to Chebyshev-Halley iterative method are introduced, including basic concepts, convergence order, convergence criterion and relevant conclusions in Banach space. The main idea of this paper and the solution process of the main thesis are given. Finally, the structure of this paper is given. In chapter 2, the convergence of a class of variant Chebyshev-Halley iterative methods is studied. The convergence criterion of iterative method and the proof of semi-local convergence are given when the conditions are satisfied. Finally, the influence of the change of parameter 伪 on the convergence radius is analyzed to provide a basis for parameter selection. First, the Chebyshev-Halley type iteration is decomposed into two parts, the first part is a simple Newton iteration, which can be solved directly according to the literature knowledge, and the second part is the difficult part, which mainly simplifies the iterative steps by using a set of monotone sequence of numbers and gives the corresponding results. In chapter 3, the convergence and iteration error of Chebyshev-Halley type iterative method are studied by cyclic sequence under the central Lipschitz condition. Then, under the condition that the first order is differentiable and the second order is nondifferentiable, the N-sorbate on method can be applied to the Chebyshev-Halley type iteration, which can reduce the computational complexity and simplify the iterative conditions theoretically.
【学位授予单位】：浙江师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.6
本文编号：2301135
[Abstract]:Modern society is a society with rapid development of information, so solving some iterative problems also needs to be synchronized with informatization. Therefore, how to increase the speed of iteration, increase the scope of iteration, and reduce the computational workload are of great importance in computational mathematics. In this paper, by changing the original criterion of iterative convergence, we extend the original range of iterative convergence, reach the convergence point quickly, and explore the semi-locally convergent iterative method. The main contents are as follows: in the first chapter, the history of Chebyshev iterative method and Halley iterative method and the preparatory knowledge related to Chebyshev-Halley iterative method are introduced, including basic concepts, convergence order, convergence criterion and relevant conclusions in Banach space. The main idea of this paper and the solution process of the main thesis are given. Finally, the structure of this paper is given. In chapter 2, the convergence of a class of variant Chebyshev-Halley iterative methods is studied. The convergence criterion of iterative method and the proof of semi-local convergence are given when the conditions are satisfied. Finally, the influence of the change of parameter 伪 on the convergence radius is analyzed to provide a basis for parameter selection. First, the Chebyshev-Halley type iteration is decomposed into two parts, the first part is a simple Newton iteration, which can be solved directly according to the literature knowledge, and the second part is the difficult part, which mainly simplifies the iterative steps by using a set of monotone sequence of numbers and gives the corresponding results. In chapter 3, the convergence and iteration error of Chebyshev-Halley type iterative method are studied by cyclic sequence under the central Lipschitz condition. Then, under the condition that the first order is differentiable and the second order is nondifferentiable, the N-sorbate on method can be applied to the Chebyshev-Halley type iteration, which can reduce the computational complexity and simplify the iterative conditions theoretically.
【学位授予单位】：浙江师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.6
【参考文献】
相关期刊论文 前1条
1 窦玮钧;Halley迭代方法简介[J];数学通报;1986年08期
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