具有可分差分的四阶广义迭代法

Samundra Regmi, I. Argyros, Gagan Deep
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引用次数: 0

摘要

来自不同学科的许多应用程序在抽象空间(例如欧几里得多维空间、希尔伯特空间或巴拿赫空间)中被表述为方程或方程组。世界各地的研究人员正在开发方法来处理这类方程的解。许多这样的方程是不可微的。这些方法也可用于求解可微方程。一种特定的方法被用作描述方法学的样本。同样的方法可以用在利用线性算子的逆的其他方法上。现有迭代方法的局部收敛问题是泰勒展开级数的使用。这样,通过假设存在迭代法中不存在的高阶导数来证明收敛性。此外,可以计算的误差距离的界限事先是不可用的。此外,也没有讨论方程解的分离性。这些问题降低了迭代方法的适用性,并构成了开发本文的动机。本文的新颖之处在于它在较弱的收敛条件下积极地解决了所有这些问题。最后,提出了利用标量数列最大化法研究收敛性的半局部分析问题。实验进一步证明了这一理论。
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Generalized Iterative Method of Order Four with Divided Differences
Numerous applications from diverse disciplines are formulated as an equation or system of equations in abstract spaces such as Euclidean multidimensional, Hilbert, or Banach, to mention a few. Researchers worldwide are developing methodologies to handle the solutions of such equations. A plethora of these equations are not differentiable. These methodologies can also be applied to solve differentiable equations. A particular method is utilized as a sample via which the methodology is described. The same methodology can be used on other methods utilizing inverses of linear operators. The problem with existing approaches on the local convergence of iterative methods is the usage of Taylor expansion series. This way, the convergence is shown but by assuming the existence of high-order derivatives which do not appear on the iterative methods. Moreover, bounds on the error distances that can be computed are not available in advance. Furthermore, the isolation of a solution of the equation is not discussed either. These concerns reduce the applicability of iterative methods and constitute the motivation for developing this article. The novelty of this article is that it positively addresses all these concerns under weaker convergence conditions. Finally, the more important and harder to study semi-local analysis of convergence is presented using majorizing scalar sequences. Experiments are further performed to demonstrate the theory.
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