一類(lèi)變異型Chebyshev-Halley迭代法的收斂性
發(fā)布時(shí)間：2018-10-30 20:46
【摘要】：現(xiàn)代的社會(huì)是信息化高速發(fā)展的社會(huì),求解一些迭代問(wèn)題同樣更要與信息化同步.因此,怎樣提高迭代速度、增加迭代范圍、減少計(jì)算工作量,都是計(jì)算數(shù)學(xué)中至關(guān)重要的.本文通過(guò)改變?cè)械牡諗颗袚?jù)、擴(kuò)大原有的迭代收斂范圍,快速地達(dá)到收斂點(diǎn),并對(duì)半局部收斂的迭代方法進(jìn)行探索.具體內(nèi)容如下：第一章主要介紹了Chebyshev迭代法和Halley迭代法的發(fā)展歷史以及與Chebyshev-Halley型迭代法相關(guān)的預(yù)備知識(shí),包括基礎(chǔ)概念、收斂階、收斂判據(jù)及Banach空間的相關(guān)結(jié)論。給出本文的主要思想及主體論文的解法過(guò)程.最后給出了本文的結(jié)構(gòu)框架.第二章研究了一類(lèi)變異型Chebyshev-Halley迭代法的收斂性.給出了在滿足條件時(shí)的迭代法收斂性判據(jù)及半局部收斂性的證明,最后分析了參數(shù)α的變化對(duì)收斂半徑的影響,以提供某種參數(shù)選擇的依據(jù).首先把Chebyshev-Halley型迭代式分解成兩部分,第一部分為簡(jiǎn)單的Newton迭代,可以直接根據(jù)文獻(xiàn)知識(shí)解決；第二部分是難點(diǎn)部分,主要利用一組單調(diào)數(shù)列簡(jiǎn)化迭代步驟并給出相應(yīng)的結(jié)果.第三章研究了在中心Lipschitz條件|下,用循環(huán)數(shù)列研究Chebyshev-Halley型迭代法的收斂性及迭代誤差.然后,進(jìn)一步探索在一階可導(dǎo)且二階不可導(dǎo)的條件下,將類(lèi)N秫on法運(yùn)用到Chebyshev-Halley型迭代中,在理論上可以減少迭代計(jì)算量并簡(jiǎn)化迭代條件.
[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 偽 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.
【學(xué)位授予單位】：浙江師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類(lèi)號(hào)】：O241.6
本文編號(hào)：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 偽 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.
【學(xué)位授予單位】：浙江師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類(lèi)號(hào)】：O241.6
【參考文獻(xiàn)】
相關(guān)期刊論文 前1條
1 竇瑋鈞;Halley迭代方法簡(jiǎn)介[J];數(shù)學(xué)通報(bào);1986年08期
，本文編號(hào)：2301135
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