A New Modified Efficient Levenberg–Marquardt Method for Solving Systems of Nonlinear Equations

Zhenxiang Wu,Yanfang Ma,Lei Li,Tong Zhou,Liang Chen,Lei Liu

doi:10.1155/2021/5608195

Abstract

For systems of nonlinear equations, a modified efficient Levenberg–Marquardt method with new LM parameters was developed by Amini et al. (2018). The convergence of the method was proved under the local error bound condition. In order to enhance this method, using nonmonotone technique, we propose a new Levenberg–Marquardt parameter in this paper. The convergence of the new Levenberg–Marquardt method is shown to be at least superlinear, and numerical experiments show that the new Levenberg–Marquardt algorithm can solve systems of nonlinear equations effectively.

Highlights

Consider the system of nonlinear equations F(x) 0, (1)where the function F(x): Rn ⟶ Rm is continuously differentiable
We say ‖F(x)‖ provides a local error bounded on N for (1), where ‖xk − xk‖ dist(xk, X∗), if there exists a positive constant C1 such that
For all x ∈ N(x∗, b) ∩ X∗, suppose that rank(J(x)) r, and we prove the local convergence of Algorithm 1 by singular value decomposition (SVD) of J(x)

Summary

Introduction

Where the function F(x): Rn ⟶ Rm is continuously differentiable. In this paper, we assume that the solution set of (1) (denoted by X∗) is nonempty, with ‖ · ‖ referring to the 2norm. In [5], Fan used λk μk‖F(xk)‖δ, δ ∈ (1, 2], where μk is updated with trust region technology in each iteration, the LM method has quadratic convergence under some suitable conditions, and λk μk‖F(xk)‖δ can alleviate the effect of the initial point being far away from the set X∗. To avoid this trouble, Amini used λk μk‖Fk‖/(1 + ‖Fk‖) in [6]; when the sequence 􏼈xk􏼉 is far from the solution set and ‖Fk‖ is very large, λk is close to μk, which effectively controls the range of λk.

The Efficient Algorithm and Global Convergence

Local Convergence

Numerical Experiments

Variably dimensioned function 10 10

Findings

Conclusion