A Self-Tuning Modified Firefly Algorithm to Solve Univariate Nonlinear Equations with Complex Roots

Abstract

The use of numerical methods to solve univariate nonlinear equations has many drawbacks. We propose a modified firefly algorithm [MOD FA] with a self-tuning ability to solve a given univariate nonlinear equation. Our modification is capable of finding almost all real as well as complex roots of a nonlinear equation within a reasonable interval/range. The modification includes an archive to collect best fireflies and a flag to determine poorly performed iterations. It is also capable of tuning the algorithm-specific parameters while finding the optimum solutions. The self-tuning concept allows the users of our application to use it without any prior knowledge of the algorithm. We validate our approach on examples of some special univariate nonlinear equations with real as well as complex roots. We have also conducted a statistical test: the Wilcockson sign rank test. By conducting a comparison with the genetic algorithm and differential evolution with same modifications [MOD GA] [MOD DE] and with the original firefly algorithm [FA], we confirm the efficiency and the accuracy of our approach.