# Local convergence analysis of the Gauss-Newton-Kurchatov method

## TL;DR

This paper analyzes the local convergence of the Gauss-Newton-Kurchatov method for nonlinear least squares problems, improving convergence region and accuracy over previous results through refined estimates and weaker hypotheses.

## Contribution

It provides an enhanced convergence analysis of the Gauss-Newton-Kurchatov method, extending the convergence region and improving solution accuracy under weaker assumptions.

## Key findings

- Extended convergence region compared to previous results
- Finer error estimates and solution localization
- Numerical examples confirm theoretical improvements

## Abstract

We present a local convergence analysis of the Gauss-Newton-Kurchatov method for solving nonlinear least squares problems with a decomposition of the operator. The method uses the sum of the derivative of the differentiable part of the operator and the divided difference of the nondifferentiable part instead of computing the full Jacobian. A theorem, which establishes the conditions of convergence, radius and the convergence order of the proposed method, is proved (Shakhno 2017). However, the radius of convergence is small in general limiting the choice of initial points. Using tighter estimates on the distances, under weaker hypotheses (Argyros et al. 2013), we provide an analysis of the Gauss-Newton-Kurchatov method with the following advantages over the corresponding results (Shakhno 2017): extended convergence region; finer error distances, and an at least as precise information on the location of the solution. The numerical examples illustrate the theoretical results.