# Towards efficient solutions: A novel approach to quadratic nonlinearity in boundary value problems

## TL;DR

This paper introduces a new iterative method for solving nonlinear equations with quadratic nonlinearity, achieving higher efficiency than the Newton method.

## Contribution

A novel single-point iterative method with a convergence order of four and an improved efficiency index for quadratic nonlinear systems.

## Key findings

- The proposed method achieves a convergence order of four with two functional evaluations.
- The efficiency index of the new method is 2, surpassing the Newton method's 1.4142.
- Numerical testing confirms the method's effectiveness on quadratic nonlinear problems.

## Abstract

The Newton method is a classical method for solving systems of nonlinear equations and offers quadratic convergence. The order of convergence of the Newton method is optimal as it requires one evaluation for the system of nonlinear equations and the second for the Jacobian. Many boundary value problems in nature have quadratic non-linearity and the corresponding system of nonlinear equations associated with their discrete formulation has constant 2nd-order Fréchet derivatives. We try to get benefit from this information and develop a single-point iterative method to solve such a system of nonlinear equations with quadratic nonlinearity. In our proposed single-point iterative method, we perform one evaluation of a system of nonlinear equations and another for Jacobian. In total, there are two functional evaluations, and we do not count the evaluation of the 2nd-order Fréchet derivative as it is constant in all the iterations of the method. The convergence order (CO) of our proposed method is four. The efficiency index of our method is 41/2 = 2 which is higher than that of the Newton method 21/2 = 1.4142. To quantify the functionality of our proposed algorithm, we have performed extensive numerical testing on a collection of test problems with quadratic nonlinearity.