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Extending the Convergence Domain of Methods of Linear Interpolation for the Solution of Nonlinear Equations
Symmetry ( IF 2.940 ) Pub Date : 2020-07-01 , DOI: 10.3390/sym12071093
Ioannis K. Argyros , Stepan Shakhno , Halyna Yarmola

Solving equations in abstract spaces is important since many problems from diverse disciplines require it. The solutions of these equations cannot be obtained in a form closed. That difficulty forces us to develop ever improving iterative methods. In this paper we improve the applicability of such methods. Our technique is very general and can be used to expand the applicability of other methods. We use two methods of linear interpolation namely the Secant as well as the Kurchatov method. The investigation of Kurchatov’s method is done under rather strict conditions. In this work, using the majorant principle of Kantorovich and our new idea of the restricted convergence domain, we present an improved semilocal convergence of these methods. We determine the quadratical order of convergence of the Kurchatov method and order 1 + 5 2 for the Secant method. We find improved a priori and a posteriori estimations of the method’s error.

中文翻译:

非线性方程解线性插值法收敛域的扩展

在抽象空间中求解方程很重要,因为来自不同学科的许多问题都需要它。这些方程的解不能以封闭的形式获得。这种困难迫使我们开发不断改进的迭代方法。在本文中,我们提高了此类方法的适用性。我们的技术非常通用,可用于扩展其他方法的适用性。我们使用两种线性插值方法,即割线法和库尔恰托夫法。库尔恰托夫方法的研究是在相当严格的条件下进行的。在这项工作中,使用 Kantorovich 的主要原理和我们的受限收敛域的新思想,我们提出了这些方法的改进的半局部收敛。我们确定了库尔恰托夫方法的二次收敛阶次和割线法的阶次 1 + 5 2。我们发现改进的方法误差的先验和后验估计。
更新日期:2020-07-01
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