# New versions of Newton method: step-size choice, convergence domain and under-determined equations

## TL;DR

This paper introduces new Newton-like algorithms with adaptive step-size choices that improve convergence properties and applicability to under-determined systems, including explicit complexity analysis and global convergence under certain conditions.

## Contribution

It proposes novel Newton-type methods with adaptive step-sizes, extending applicability to under-determined systems and providing convergence domain estimates and complexity results.

## Key findings

- New algorithms with improved convergence domains.
- Global convergence under specific assumptions.
- Explicit complexity bounds for the proposed methods.

## Abstract

Newton method is one of the most powerful methods for finding solutions of nonlinear equations and for proving their existence. In its "pure" form it has fast convergence near the solution, but small convergence domain. On the other hand damped Newton method has slower convergence rate, but weaker conditions on the initial point. We provide new versions of Newton-like algorithms, resulting in combinations of Newton and damped Newton method with special step-size choice, and estimate its convergence domain. Under some assumptions the convergence is global. Explicit complexity results are also addressed. The adaptive version of the algorithm (with no a priori constants knowledge) is presented. The method is applicable for under-determined equations (with $m<n$, $m$ being the number of equations and $n$ being the number of variables). The results are specified for systems of quadratic equations, for composite mappings and for one-dimensional equations and inequalities.