Shamanskii method for solving parameterized fuzzy nonlinear equations

Abstract

One of the most significant problems in fuzzy set theory is solving fuzzy nonlinearequations. Numerous researches have been done on numerical methods for solving these problems, but numerical investigation indicates that most of the methods arecomputationally expensive due to computing and storage of Jacobian orapproximate Jacobian at every iteration. This paper presents the Shamanskiialgorithm, a variant of Newton method for solving nonlinear equation with fuzzyvariables. The algorithm begins with Newton’s step at first iteration, followed byseveral Chord steps thereby reducing the high cost of Jacobian or approximateJacobian evaluation during the iteration process. The fuzzy coefficients of thenonlinear systems are parameterized before applying the proposed algorithm toobtain their solutions. Preliminary results of some benchmark problems andcomparisons with existing methods show that the proposed method is promising.