Visual Analysis of Dynamics Behaviour of an Iterative Method Depending on Selected Parameters and Modifications

Abstract

<jats:p>There is a huge group of algorithms described in the literature that iteratively find solutions of a given equation. Most of them require tuning. The article presents root-finding algorithms that are based on the Newton–Raphson method which iteratively finds the solutions, and require tuning. The modification of the algorithm implements the best position of particle similarly to the particle swarm optimisation algorithms. The proposed approach allows visualising the impact of the algorithm’s elements on the complex behaviour of the algorithm. Moreover, instead of the standard Picard iteration, various feedback iteration processes are used in this research. Presented examples and the conducted discussion on the algorithm’s operation allow to understand the influence of the proposed modifications on the algorithm’s behaviour. Understanding the impact of the proposed modification on the algorithm’s operation can be helpful in using it in other algorithms. The obtained images also have potential artistic applications.</jats:p>

A New Root‐Finding Algorithm for Solving Real‐World Problems and Its Complex Dynamics via Computer Technology

“…For further details about polynomiography and its applications, see [26][27][28][29][30][31][32][33][34][35][36][37][38] and the references therein.…”

Numerical Methods With Engineering Applications and Their Visual Analysis via Polynomiography

“…A plethora of such type of images can be drawn through the computer program by the variation of k which denotes the upper bound of iterations consumed by the iteration scheme. For further study in the field of polynomiography along with the artistic applications, one can see [6], [7], [9], [11], [13], [16]- [18], [26], [34] and the references therein.…”

“…The term "polynomiography"was introduced to science in 21 st century by Kalantari [15], that was the last thought-provoking contribution to the polynomials root-finding history. He further described polynomiography as a process that creates aesthetically pleasing and beautiful graphics, it was patented by Kalantari in USA in 2005 [9], [16]. An individual image which is generated in the process of polynomiography, is known as a "polynomiograph".…”