A Novel Root-Finding Algorithm With Engineering Applications and its Dynamics via Computer Technology

Abstract

Root-finding of non-linear equations is one of the most appearing problems in engineering sciences. Most of the complicated engineering problems can be modeled easily by means of non-linear functions. The role of iterative algorithms via computers for solving such functions is much important and cannot be denied in the modern age. In an iterative algorithm, the convergence order and the computational cost per iteration are the main characteristics that depict its efficiency and performance i.e., a method with higher-order and lower computational cost will be more efficient and vice versa. Keeping these facts into consideration, the main goal of this paper is to introduce a new derivative-free iterative method that performs better. We develop this algorithm by utilizing the forward-and finite-difference schemes on well-known Househ"older's method, resulting in an efficient and derivative-free algorithm with a low per iteration computing cost. We also look at the developed algorithm's convergence criterion and show that it is quartic-order convergent. We investigate nine test-examples and solve them to demonstrate its correctness, validity, and efficiency numerically. Some real-world engineering problems in civil and chemical engineering are also included in these examples. The numerical results of the test-examples reveal that the newly constructed method outperforms the existing similar algorithms found in the literature. We consider various different-degrees complex polynomials for the graphical analysis and used a computer tool to create the polynomiographs of the proposed quartic-order algorithm and compare it to other comparable existing approaches. The graphical findings show that the developed method has a faster convergence speed than the other comparable algorithms.

A Novel Multistep Iterative Technique for Models in Medical Sciences with Complex Dynamics

“…Its computational process is represented below, and it involves two function evaluations: one for the function itself and another for the first-order derivative of the function where ζ ′( x n ) ≠ 0. In [ 19 ], researchers developed a modified version of an existing approach aimed at reducing first-order derivatives. They devised a two-step approach with fourth-order convergence, denoted as N 4.…”

“…where ζ′ðx n Þ ≠ 0. In [19], researchers developed a modified version of an existing approach aimed at reducing firstorder derivatives. They devised a two-step approach with fourth-order convergence, denoted as N4.…”

“…Substituting ( 13), ( 14), ( 17), (18), and (19) in the third step of ( 7), one obtains Finally, using Equations ( 12) and ( 16), we obtain the required results as follows:…”

Novel Parametric Families of with and without Memory Iterative Methods for Multiple Roots of Nonlinear Equations

“…Equation ( 2) has a quadratic order of convergence for m ≥ 1. While there exist numerous with and without memory iterative methods for computing simple roots [1][2][3][4][5][6], the task of developing efficient methods for finding multiple roots remains challenging. Despite the availability of several without memory iterative methods for multiple roots (see [7][8][9][10] and the references therein), there is a paucity of methods that utilise multiple points with memory and accelerating parameters for computing multiple roots.…”

Modified Householder Method of Fifth Order of Convergence and Its Dynamics on Complex Plane

“…The introduction of higher order derivative often presents challenges including increased computational costs and practical application difficulties. Consequently, many researchers seek to find free second derivatives iterative methods with various approaches as evidenced in works such as [3][4][5][6][7][8].…”