Extending the Convergence Domain of Methods of Linear Interpolation for the Solution of Nonlinear Equations

• Main Authors: Ioannis K. Argyros, Stepan Shakhno, Halyna Yarmola
• Format: Article
• Language: English
• Published: MDPI AG 2020-07-01
• Series: Symmetry
• Subjects: nonlinear equation; iterative process; convergence order; secant method; Kurchatov method; Banach space
• Online Access: https://www.mdpi.com/2073-8994/12/7/1093

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 <inline-formula> <math display="inline"> <semantics> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>5</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </semantics> </math> </inline-formula> for the Secant method. We find improved a priori and a posteriori estimations of the method’s error.