Solving System of Nonlinear Equations Using Methods in the Halley Class

Abstract

In this thesis a new iterative frame work to solve the nonlinear system of equations \(F(x)=0\) in n-dimensional real space is established. This iterative frame work is based on a quadratic model of the function \(F(x)\) at the current point. The convergence analysis shows that this frame work has Q-third rate of convergence. The main advantages of this frame work that the system of nonlinear equations simplified to quadratic system of equations which hopefully has less computational complexity than the original system. It is shown that the Halley class inherits it's convergence properties from the quadratic model. In practice, for the large-scale problems, the inexact Halley class methods is used to solve \(F(x)=0\), and has Q-third rate of convergence.