# The Combination of Bisection Method and Artificial Bee Colony Algorithm for Solving Hard Fix Point Problems

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 188)

## Abstract

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.

### Keywords

Bisection method Root-finding method Fix point problems Artificial Bee Colony algorithm

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