A self-tuning algorithm to approximate roots of systems of nonlinear equations based on the firefly algorithm

Abstract

The most acquainted methods to find root approximations of nonlinear equations and systems; numerical methods possess disadvantages such as necessity of acceptable initial guesses and the differentiability of the functions. Even having such qualities, for some univariate nonlinear equations and systems, approximations of roots is not possible with numerical methods. Research are geared towards finding alternate approaches, which are successful where numerical methods fail. One of the most disadvantageous properties in such approaches is inability of finding more than one approximation at a time. On the other hand these methods are incorporated with algorithm specific parameters which should be set properly in order to achieve good results. We present a modified firefly algorithm handling the problem as an optimisation problem, which is capable of giving multiple root approximations simultaneously within a reasonable state space while tuning the parameters of the proposed algorithm by itself, using a self-tuning framework. Differentiability and the continuity of the functions and the close initial guesses are needless to solve nonlinear systems using the proposed approach. Benchmark systems found in the literature were used to test the new algorithm. The root approximations and the tuned parameters obtained along with the statistical analysis illustrate the viability of the method.