Shamanskii Method for Solving Parameterized Fuzzy Nonlinear Equations

Abstract

One of the most significant problems in fuzzy set theory is solving fuzzy nonlinear equations. Numerous researches have been done on numerical methods for solving these problems, but numerical investigation indicates that most of the methods are computationally expensive due to computing and storage of Jacobian or approximate Jacobian at every iteration. This paper presents the Shamanskii algorithm, a variant of Newton method for solving nonlinear equation with fuzzy variables. The algorithm begins with Newton’s step at first iteration, followed by several Chord steps thereby reducing the high cost of Jacobian or approximate Jacobian evaluation during the iteration process. The fuzzy coe?cients of the nonlinear systems are parameterized before applying the proposed algorithm to obtain their solutions. Preliminary results of some benchmark problems and comparisons with existing methods show that the proposed method is promising.