Performance analysis of a modified Newton method for parameterized dual fuzzy nonlinear equations and its application

Abstract

In this paper, we are interested in the numerical solution of nonlinear equations, particularly, when the coefficients are fuzzy numbers rather than crisp numbers. These problems often arise in numerous applications and have recently been the subject of several studies, where many researchers parameterized the fuzzy coefficients and proposed numerical method for obtaining the solutions. Most of the numerical methods applied for solving the parameterized fuzzy equations are Newton-like methods that require the computation of Jacobian matrix at every iteration or after every few iterations. However, the Newton search direction may not be a descent direction if the Jacobian J ( x k ) is not positive definite, and, when J ( x k ) is singular, the method may fail to converge. Also, there are limited literatures on numerical methods for solution of dual fuzzy nonlinear equations. This paper proposed a variant of Newton’s method for solving parameterized dual fuzzy nonlinear equations. This method introduces a new parameter μ k to the Newton’s method to ensure that the search direction is a descent direction even when J ( x k ) is not positive definite. The proposed method was further applied to solve an application problem. Preliminary results on the considered benchmark and application problems show that the new algorithm effective and promising.

Keywords

• Nonlinear equation
• Fuzzy coefficient
• Parametric form
• Newton method
• Jacobian