AT algorithm for fractal polynomiographs: a study on fast convergence under weak contraction mappings

Abstract

In the present work, we use the AT algorithm, an iteration consisting of three steps that approximates the fixed point of a weak contraction. The algorithm not only demonstrates faster convergence compared to established methods such as the S, Normal-S, Varat, Mann, Ishikawa, and Picard iterations for weak contraction, but also exhibits strong convergence properties. The paper also explores the AT algorithm’s almost stable behavior for weak contraction. We further apply the AT iterative scheme to construct Julia sets and polynomiographs, providing a practical comparison with the AET values for the Normal-S, Mann, Picard, and AT iterations, thereby demonstrating the real-world relevance of our research.