解非线性方程的高阶迭代算法及其收敛性分析
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摘要
非线性问题一直是近代数学研究的主流之一,而迭代法是求解Banach空间中非线性方程F(x)=0问题的最有效的方法。随着数学研究本身的发展和大型计算机的出现及完善,各种非线性问题日益引起科学家和工程技术人员的兴趣和重视。特别是有关近代物理和科学工程计算中的一些关键问题,归根结底都依赖于某些特定的非线性方程的求解。而迭代法的优劣对于非线性问题的求解速度的快慢和结果的好坏有很大的影响。近几十年来,计算机的迅猛发展有力地推动着数值分析的研究工作。一些经典的方法经过严格的实践检验后,显露出了若干缺陷,而这些缺陷在碰到计算量非常大的实际问题时,显得尤为突出。在大规模计算中,计算效率至关重要,人们往往对不同的问题选择不同的算法,以尽可能的避免使用低效率的算法。因此,我们在考虑算法收敛阶的同时,对算法在计算过程中每一步的计算量也尤为关心。所以从实际出发,进行具有高计算效能迭代算法的研究有重要的科学价值和实际意义。
全文共分四部分。第一章概述了迭代法几个世纪的发展情况,介绍了一些具有代表性的迭代方法以及相关的迭代法的基本理论。近几十年来,数值工作者们不断的提出一些新的迭代格式,事实上这些新方法大多是根据实际情况的需要对经典的迭代格式进行修正和变形,因此Newton法等一系列经典的迭代法就成为我们讨论新的迭代方法的起点。数学家们对这些方法都做了很深入的研究,关于这方面的文章著作也是数不胜数,其中有非常丰富的理论结果和证明技巧是可以借鉴的。
第二章提出了一族求解非线性方程的高阶收敛的迭代方法,并用基本的数学分析的方法对其收敛性进行了分析和证明。这里所说的高阶收敛不同于在第一章里提到的三阶、四阶等具有固定收敛阶的高阶方法,而是指迭代法的具体形式和收敛阶数都会随着条件的变化而变化的。从理论上来讲,只要条件具备,这类型的迭代法是可以达到任意阶收敛的。一般来说,迭代法的收敛阶越高,条件和形式相应也会越复杂。比如Euler迭代族和Halley迭代族,这两族方法是可以达到高阶收敛的,但也必须计算高阶导数值。这一章里我们给出的新方法在迭代过程中不需要计算函数的高阶(二阶或二阶以上)导数值,只需要计算函数的一阶导数值,就可以达到较高的收敛阶。相比之下,我们的方法在达到相同收敛阶的同时,计算复杂性明显降低。尤其是在多维空间下求解的时候,就会有更明显的优势。另外我们还给出了一些具体的数值例子来进一步说明此方法在不需要计算高阶导数的情况下,同样可以达到很快的收敛速度。
第三章我们通过将Newton法与其它迭代法组合,得到了两族新的迭代法。本章的重点是介绍其构造方法和讨论其收敛性问题。所谓的组合就是用两个相同的或不同的迭代法构造出一个新的迭代法,这个新的迭代法里综合利用了原迭代法中的函数和导数的信息。两个迭代法组合后,其收敛阶自然也会相应有所提高,但是计算代价也可能会相应增加不少。而本章所构造的组合迭代法只需要多计算一个函数值,就可以使收敛阶在原迭代法的基础上提高了λ(1＜λ≤2)阶,同时还避免了计算二阶或二阶以上的导数值的麻烦。接着,我们运用这一组合方法构造出几个具体的迭代法,并通过计算一些数值实例和其它迭代法进行比较,不难发现这类组合方法比起一些要求相同计算代价的迭代法,有更高的计算效率。
第四章介绍了一个变形的Jarratt方法。变形后的迭代法可以避免计算导数值的逆。我们用优序列的技巧在kantorovich条件下对这一变形的迭代法的收敛性分析进行了讨论,给出了其半局部收敛性定理。此变形的迭代法在迭代过程中不需要计算任何导数的逆,这在实际问题的计算中,可以大大提高计算效率。
Nonlinear problems is one of the main fields in modern mathematical research. The iterative method is an excellent method to solve nonlinear operator equations f(x) = 0 in Banach space. Following the development in research of Mathematics and the appearance and consummation of mainframe computer, several kinds of nonlinear problems have been regarded by many scientists and engineers sooner or later. They have more and more interest in these questions, especially, some key questions in engineering calculation of recent physics and science are depended on the solution of nonlinear equation. In numerical scientific computation, especially for complicated scientific and engineering problems, whether the nonlinear problems will be solved well or not is directly affected by the choice of iterative methods. So it is very important and meaningful to do the research of iterative methods. For the past several decades, the rapid advancement of computers has effectively promoted the research on numerical analysis. Strictly examined by practices, some classical methods are proved to have some drawbacks which remain especially evident in the solving of some practical problems with mass calculational scales. The computational efficiency plays a crucial role in large scale calculation, which calls on us to avoid applying computational methods with low efficiency. Thus, while considering the convergence order of computational method, we also take computational cost in every step into consideration particularly. Therefore, the research on iterative method with high efficiency means a lot in terms of both scientific research and practical application.
This paper consists of four chapters. In chapter 1, we summarize mainly about the development of iterative methods during the past several centuries and introduce several representative iterative formulae and some basic theories on relevant iterative methods. For the past several decades, professional numerical researchers have kept coming up with various new iterative methods, most of which, in fact, are the modification and deformation of classical iterative method according actual necessaries. Thus, a series of classical iterative methods such as the Newton method and so on is the starting point in our discussion on the new methods. Mathematicians have conducted profound researches in this regard, producing large numbers of papers and literatures, from which we have profuse theoretical fruit and skills to borrow.
In the second chapter, we present a family of iterative methods with higher-order convergence for solving nonlinear equations, and analysis and prove the convergence by basic method of mathematical analysis. The higher-order convergence we discussed here is different from third or fourth order iterative method with fixed convergence order in the opening chapter, but refers to those whose concrete formula and convergence order change under different circumstances. Theoretically, the more complex the condition is, the more complex the form is, and the higher the convergence is. Take Euler and Halley iterative methods as examples, they have high convergence order, while entailing the calculation of derivative of higher order. The new method we put forward here which needn't evaluate second or higher derivative of the function but first derivative can have higher-order convergence. Compared with other methods, our has lower computational complexity. In particular, it presents an obvious advantage regarding the problems on n-dimensions space. Besides, we display some examples further indicating that our method can have higher convergence rate without the computation of the higher-order derivative of the function.
In the third chapter, we combine the Newton and other iterative methods, producing two families of new iterative methods. Here, we concern our efforts on the introduction of its construction technique and discussion on its convergence. Combination means that we formulate a new iterative method using two same or different iterative method. Generally speaking, the convergence of combined iterative method can be improved, but its computational cost also increases accordingly. However, the convergence order of the combined iterative method we produced here through only adding one evaluation of the function at another iterated point. Then we use this combined technique to construct several actual iterative methods, and compare them with other methods by applying them in some calculational problems, indicating that our methods has higher computa- tional efficiency.
In the fourth chapter, we introduce a deformed Jarratt method. We needn't computing the inverse of first derivative in the new method. Using the majorant function, we discuss the convergence analysis of the deformed method under the condition of Kantorovich, and come up with convergence theorem. We also prove that the new method need not calculate any inverse of derivative value. Applied in the practical problems, this can promote the efficiency greatly.
全文共分四部分。第一章概述了迭代法几个世纪的发展情况,介绍了一些具有代表性的迭代方法以及相关的迭代法的基本理论。近几十年来,数值工作者们不断的提出一些新的迭代格式,事实上这些新方法大多是根据实际情况的需要对经典的迭代格式进行修正和变形,因此Newton法等一系列经典的迭代法就成为我们讨论新的迭代方法的起点。数学家们对这些方法都做了很深入的研究,关于这方面的文章著作也是数不胜数,其中有非常丰富的理论结果和证明技巧是可以借鉴的。
第二章提出了一族求解非线性方程的高阶收敛的迭代方法,并用基本的数学分析的方法对其收敛性进行了分析和证明。这里所说的高阶收敛不同于在第一章里提到的三阶、四阶等具有固定收敛阶的高阶方法,而是指迭代法的具体形式和收敛阶数都会随着条件的变化而变化的。从理论上来讲,只要条件具备,这类型的迭代法是可以达到任意阶收敛的。一般来说,迭代法的收敛阶越高,条件和形式相应也会越复杂。比如Euler迭代族和Halley迭代族,这两族方法是可以达到高阶收敛的,但也必须计算高阶导数值。这一章里我们给出的新方法在迭代过程中不需要计算函数的高阶(二阶或二阶以上)导数值,只需要计算函数的一阶导数值,就可以达到较高的收敛阶。相比之下,我们的方法在达到相同收敛阶的同时,计算复杂性明显降低。尤其是在多维空间下求解的时候,就会有更明显的优势。另外我们还给出了一些具体的数值例子来进一步说明此方法在不需要计算高阶导数的情况下,同样可以达到很快的收敛速度。
第三章我们通过将Newton法与其它迭代法组合,得到了两族新的迭代法。本章的重点是介绍其构造方法和讨论其收敛性问题。所谓的组合就是用两个相同的或不同的迭代法构造出一个新的迭代法,这个新的迭代法里综合利用了原迭代法中的函数和导数的信息。两个迭代法组合后,其收敛阶自然也会相应有所提高,但是计算代价也可能会相应增加不少。而本章所构造的组合迭代法只需要多计算一个函数值,就可以使收敛阶在原迭代法的基础上提高了λ(1＜λ≤2)阶,同时还避免了计算二阶或二阶以上的导数值的麻烦。接着,我们运用这一组合方法构造出几个具体的迭代法,并通过计算一些数值实例和其它迭代法进行比较,不难发现这类组合方法比起一些要求相同计算代价的迭代法,有更高的计算效率。
第四章介绍了一个变形的Jarratt方法。变形后的迭代法可以避免计算导数值的逆。我们用优序列的技巧在kantorovich条件下对这一变形的迭代法的收敛性分析进行了讨论,给出了其半局部收敛性定理。此变形的迭代法在迭代过程中不需要计算任何导数的逆,这在实际问题的计算中,可以大大提高计算效率。
Nonlinear problems is one of the main fields in modern mathematical research. The iterative method is an excellent method to solve nonlinear operator equations f(x) = 0 in Banach space. Following the development in research of Mathematics and the appearance and consummation of mainframe computer, several kinds of nonlinear problems have been regarded by many scientists and engineers sooner or later. They have more and more interest in these questions, especially, some key questions in engineering calculation of recent physics and science are depended on the solution of nonlinear equation. In numerical scientific computation, especially for complicated scientific and engineering problems, whether the nonlinear problems will be solved well or not is directly affected by the choice of iterative methods. So it is very important and meaningful to do the research of iterative methods. For the past several decades, the rapid advancement of computers has effectively promoted the research on numerical analysis. Strictly examined by practices, some classical methods are proved to have some drawbacks which remain especially evident in the solving of some practical problems with mass calculational scales. The computational efficiency plays a crucial role in large scale calculation, which calls on us to avoid applying computational methods with low efficiency. Thus, while considering the convergence order of computational method, we also take computational cost in every step into consideration particularly. Therefore, the research on iterative method with high efficiency means a lot in terms of both scientific research and practical application.
This paper consists of four chapters. In chapter 1, we summarize mainly about the development of iterative methods during the past several centuries and introduce several representative iterative formulae and some basic theories on relevant iterative methods. For the past several decades, professional numerical researchers have kept coming up with various new iterative methods, most of which, in fact, are the modification and deformation of classical iterative method according actual necessaries. Thus, a series of classical iterative methods such as the Newton method and so on is the starting point in our discussion on the new methods. Mathematicians have conducted profound researches in this regard, producing large numbers of papers and literatures, from which we have profuse theoretical fruit and skills to borrow.
In the second chapter, we present a family of iterative methods with higher-order convergence for solving nonlinear equations, and analysis and prove the convergence by basic method of mathematical analysis. The higher-order convergence we discussed here is different from third or fourth order iterative method with fixed convergence order in the opening chapter, but refers to those whose concrete formula and convergence order change under different circumstances. Theoretically, the more complex the condition is, the more complex the form is, and the higher the convergence is. Take Euler and Halley iterative methods as examples, they have high convergence order, while entailing the calculation of derivative of higher order. The new method we put forward here which needn't evaluate second or higher derivative of the function but first derivative can have higher-order convergence. Compared with other methods, our has lower computational complexity. In particular, it presents an obvious advantage regarding the problems on n-dimensions space. Besides, we display some examples further indicating that our method can have higher convergence rate without the computation of the higher-order derivative of the function.
In the third chapter, we combine the Newton and other iterative methods, producing two families of new iterative methods. Here, we concern our efforts on the introduction of its construction technique and discussion on its convergence. Combination means that we formulate a new iterative method using two same or different iterative method. Generally speaking, the convergence of combined iterative method can be improved, but its computational cost also increases accordingly. However, the convergence order of the combined iterative method we produced here through only adding one evaluation of the function at another iterated point. Then we use this combined technique to construct several actual iterative methods, and compare them with other methods by applying them in some calculational problems, indicating that our methods has higher computa- tional efficiency.
In the fourth chapter, we introduce a deformed Jarratt method. We needn't computing the inverse of first derivative in the new method. Using the majorant function, we discuss the convergence analysis of the deformed method under the condition of Kantorovich, and come up with convergence theorem. We also prove that the new method need not calculate any inverse of derivative value. Applied in the practical problems, this can promote the efficiency greatly.
引文
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