The Combination of Bisection Method and Artificial Bee Colony Algorithm for Solving Hard Fix Point Problems
In this work, with combination Bisection method and Artificial bee colony algorithm together(BIS-ABC), We introduce the novel iteration method to solve the real-valued fix point problem f(x) = x, x ∈ [a, b] ⊆ R. Let f(a).f(b) < 0 and there exist α ∈ [a, b], f(α) = α. In this way, without computing derivative of function f, real-roots determined with this method that is faster than ABC algorithm. In numerical analysis, Newton-Raphson method is a method for finding successively better approximations to the simple real roots, if f′(x i ) – 1 = 0, in i th iteration, then Newton’s method will terminate and we need to change initial value of a root and do algorithm again to obtain better approximate of α. But in proposed method, we reach to solution with direct search in [a, b], that includes α(convergence speed maybe less than of Newton’s method). We illustrate this method by offering some numerical examples and compare results with ABC algorithm.
KeywordsBisection method Root-finding method Fix point problems Artificial Bee Colony algorithm
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