Asymptotically Newton-Type Methods without Inverses for Solving Equations

Ioannis K. Argyros (), Santhosh George, Stepan Shakhno, Samundra Regmi, Mykhailo Havdiak and Michael I. Argyros
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Ioannis K. Argyros: Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA
Santhosh George: Department of Mathematical & Computational Science, National Institute of Technology Karnataka, Surathkal 575025, India
Stepan Shakhno: Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, Universytetska Str. 1, 79000 Lviv, Ukraine
Samundra Regmi: Department of Mathematics, University of Houston, Houston, TX 77205, USA
Mykhailo Havdiak: Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, Universytetska Str. 1, 79000 Lviv, Ukraine
Michael I. Argyros: Department of Computer Science, University of Oklahoma, Norman, OK 73501, USA

Mathematics, 2024, vol. 12, issue 7, 1-19

Abstract: The implementation of Newton’s method for solving nonlinear equations in abstract domains requires the inversion of a linear operator at each step. Such an inversion may be computationally very expensive or impossible to find. That is why alternative iterative methods are developed in this article that require no inversion or only one inversion of a linear operator at each step. The inverse of the operator is replaced by a frozen sum of linear operators depending on the Fréchet derivative of an operator. The numerical examples illustrate that for all practical purposes, the new methods are as effective as Newton’s but much cheaper to implement. The same methodology can be used to create similar alternatives to other methods using inversions of linear operators such as divided differences or other linear operators.

Keywords: Newton-type method; banach space; frozen sum of operators; Fréchet derivative; convergence; inverse of a linear operator (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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