An Accelerated Scheme for Solving Parameterized Fuzzy Nonlinear Equations

Abstract

One of the most momentous problems in fuzzy set theory is solving fuzzy nonlinear equations. Several studies have applied different numerical methods to solve this problem focusing on equations with nonsingular Jacobian at the solution point. Numerical investigation indicates that most of these proposed methods are computationally expensive due to the storage of Jacobian or approximate Jacobian at every iteration. Also, the algorithms may not be defined when the Jacobian is singular.  This paper presents a new algorithm for solving fuzzy nonlinear equation (FNE). Our algorithm introduced a new Jacobian updating formula to the Shamanskii method and thus update the Jacobian once though out the iterations unlike in Newton-type methods which compute the Jacobian matrix in every iteration. The expectation has been to reduce the computational cost at every iteration as in other numerical methods. The fuzzy quantities are presented in its parametrized form.  Numerical results have been presented which shows the methods is efficient and promising.