Converging efficiency: Computational and fractal insights into parallel non-linear schemes

Abstract

This study presents and examines a parallel method for the simultaneous approximation of all roots of nonlinear equations. With the use of a parallel computing architecture, the algorithm aims to enhance computational efficiency. An exhaustive convergence analysis corroborates the finding that the developed scheme converges at sixth order. To optimize parameter values and speed up the convergence rate of the proposed parallel technique, the concepts of dynamical and parametric planes are employed. The computational efficiency percentage demonstrates that the new parallel method is more efficient and involves fewer arithmetic operations compared to the current methods. Randomly chosen initial values are employed to demonstrate the engineering problems have been subjected to comparative analysis, which shows that the suggested parallel schemes surpass traditional methods in residual error, convergence rate, CPU time, memory usage, and computational cost. The findings indicate that the approach holds promise as a means of addressing nonlinear equations in scientific and engineering contexts.