A Two-Step Newton-Type Method for Solving System of Absolute Value Equations

Lei Shi,Javed Iqbal,Alamgir Khan,Muhammad Arif,Mahmoud Mesbah

doi:10.1155/2020/2798080

Abstract

In this paper, we suggest a Newton-type method for solving the system of absolute value equations. This new method is a two-step method with the generalized Newton method as predictor. Convergence of the proposed method is proved under some suitable conditions. At the end, we take several numerical examples to show that the new method is very effective.

Highlights

We consider the system of absolute value equations of the form: Ax − |x| b, (1)where A ∈ Rn×n and b ∈ Rn are known vectors and |x| denotes the absolute values of the components of x ∈ Rn. e system of absolute value equations arises in optimization, the economies with institutional restrictions upon prices, the free boundary problems for journal bearing lubrication, and the network equilibrium problems, for example, see [1,2,3,4,5,6,7,8,9,10,11,12]
We present a Newton-type method for solving the system of absolute value equations. e new method is a two-step method where the well-known numerical quadrature technique is used in the corrector step
Numerical results prove that the Newton-type method is very effective for solving large systems

Summary

Introduction

Where A ∈ Rn×n and b ∈ Rn are known vectors and |x| denotes the absolute values of the components of x ∈ Rn. e system of absolute value equations arises in optimization, the economies with institutional restrictions upon prices, the free boundary problems for journal bearing lubrication, and the network equilibrium problems, for example, see [1,2,3,4,5,6,7,8,9,10,11,12]. Mansoori and Erfanian [13] suggested a dynamic model to obtain the exact solution of equation (1). E development of multistep methods has gained popularity in the field of computational mathematics. Several multistep methods are proposed to solve equation (1), for example, see [3,4,5,6]. We present a Newton-type method for solving the system of absolute value equations. E Newton-type method is very simple and easy to implement in practice. E existence and uniqueness of solution shows the importance of the suggested method. Numerical results prove that the Newton-type method is very effective for solving large systems.

Proposed Method

Numerical Results

Method n

Conclusion