Some New Iterative Algorithms for Solving One-Dimensional Non-Linear Equations and Their Graphical Representation

Naseem, Amir; Rehman, M. A.; Abdeljawad, Thabet

doi:10.1109/access.2021.3049428

Cited by 17 publications

(11 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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.…”

Section: Complex Dynamicsmentioning

confidence: 99%

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

2021

Self Cite

View full text Add to dashboard Cite

Nowadays, the use of computers is becoming very important in various fields of mathematics and engineering sciences. Many complex statistics can be sorted out easily with the help of different computer programs in seconds, especially in computational and applied Mathematics. With the help of different computer tools and languages, a variety of iterative algorithms can be operated in computers for solving different nonlinear problems. The most important factor of an iterative algorithm is its efficiency that relies upon the convergence rate and computational cost per iteration. Taking these facts into account, this article aims to design a new iterative algorithm that is derivative-free and performs better. We construct this algorithm by applying the forward- and finite-difference schemes on Golbabai–Javidi’s method which yields us an efficient and derivative-free algorithm whose computational cost is low as per iteration. We also study the convergence criterion of the designed algorithm and prove its quartic-order convergence. To analyze it numerically, we consider nine different types of numerical test examples and solve them for demonstrating its accuracy, validity, and applicability. The considered problems also involve some real-life applications of civil and chemical engineering. The obtained numerical results of the test examples show that the newly designed algorithm is working better against the other similar algorithms in the literature. For the graphical analysis, we consider some different degrees’ complex polynomials and draw the polynomiographs of the designed quartic-order algorithm and compare it with the other similar existing methods with the help of a computer program. The graphical results reveal the better convergence speed and the other graphical characteristics of the designed algorithm over the other comparable ones.

show abstract

“…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.…”

Section: Complex Dynamicsmentioning

confidence: 99%

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

2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…A variety of different color graphical objects can be plotted by changing the parameters ε, L and the iteration scheme. For further details about polynomiography and its applications, see [28]- [40] and the references are cited therein. We take the following four different complex polynomials for plotting graphical objects in the complex plane: For coloring the iterations, we employ the colormap given in Figure 1.…”

Section: Polynomiographymentioning

confidence: 99%

Real-World Applications of a Newly Designed Root-Finding Algorithm and Its Polynomiography

Naseem¹,

Rehman

Abdeljawad

2021

IEEE Access

Self Cite

View full text Add to dashboard Cite

Solving non-linear equations in different scientific disciplines is one of the most important and frequently appearing problems. A variety of real-world problems in different scientific fields can be modeled via non-linear equations. Iterative algorithms play a vital role in finding the solution of such nonlinear problems. This article aims to design a new iterative algorithm that is derivative-free and performing better. We construct this algorithm by applying the forward-and finite-difference schemes on the wellknown Traubs's method which yields us an efficient and derivative-free algorithm whose computational cost is low as per iteration. We also study the convergence criterion of the designed algorithm and prove its fifth-order convergence. To demonstrate the accuracy, validity and applicability of the designed algorithm, we consider eleven different types of numerical test examples and solve them. The considered problems also involve some real-life applications of civil and chemical engineering. The obtained numerical results of the test examples show that the newly designed algorithm is working better against the other similarorder algorithms in the literature. For the graphical analysis, we consider some different-degree complex polynomials and draw polynomiographs of the designed fifth-order algorithm and compare them with the other fifth-order methods with the help of a computer program Mathematica 12.0. The graphical results show the convergence speed and other graphical characteristics of the designed algorithm and prove its supremacy over the other comparable ones.

show abstract

“…Over the time, many experts worked on iterative algorithms and brought several modified versions of Newton's algorithm with higher-order convergence which involve predictor and corrector steps and are often referred to as multistep iterative algorithms. For more information, one can see [9][10][11][12][13][14][15][16][17][18][19][20] and the references are cited therein. In general, the convergence order of a multistep algorithm is higher because of predictor and corrector steps, but it results in a higher computational cost which is the downside of these algorithms.…”

Section: Introductionmentioning

confidence: 99%

Some Real‐Life Applications of a Newly Designed Algorithm for Nonlinear Equations and Its Dynamics via Computer Tools

2021

Self Cite

View full text Add to dashboard Cite

In this article, we design a novel fourth-order and derivative free root-finding algorithm. We construct this algorithm by applying the finite difference scheme on the well-known Ostrowski’s method. The convergence analysis shows that the newly designed algorithm possesses fourth-order convergence. To demonstrate the applicability of the designed algorithm, we consider five real-life engineering problems in the form of nonlinear scalar functions and then solve them via computer tools. The numerical results show that the new algorithm outperforms the other fourth-order comparable algorithms in the literature in terms of performance, applicability, and efficiency. Finally, we present the dynamics of the designed algorithm via computer tools by examining certain complex polynomials that depict the convergence and other graphical features of the designed algorithm.

show abstract

Some New Iterative Algorithms for Solving One-Dimensional Non-Linear Equations and Their Graphical Representation

Cited by 17 publications

References 32 publications

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

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

Real-World Applications of a Newly Designed Root-Finding Algorithm and Its Polynomiography

Some Real‐Life Applications of a Newly Designed Algorithm for Nonlinear Equations and Its Dynamics via Computer Tools

Contact Info

Product

Resources

About