A Novel Root-Finding Algorithm With Engineering Applications and its Dynamics via Computer Technology (original) (raw)
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IEEE Access
The primary objective of this paper is to develop a new method for root-finding by combining forward and finite-difference techniques in order to provide an efficient, derivative-free algorithm with a lower processing cost per iteration. This will be accomplished by combining forward and finite-difference techniques. We also detail the convergence criterion that was devised for the root-finding approach, and we show that the method that was recommended is quintic-order convergent. We addressed a few engineering issues in order to illustrate the validity and application of the developed root-finding algorithm. The quantitative results justified the constructed root-finding algorithm's robust performance in comparison to other quintic-order methods that can be found in the literature. For the graphical analysis, we make use of the newly discovered method to plot some novel polynomiographs that are attractive to the eye, and then we evaluate these new plots in relation to previously established quintic-order root-finding strategies. The graphic analysis demonstrates that the newly created method for root-finding has better convergence with the larger area than the other comparable methods do.
Real-World Applications of a Newly Designed Root-Finding Algorithm and Its Polynomiography
IEEE Access, 2021
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.
Complexity, 2021
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 algor...
Root Finding With Some Engineering Applications
2016
In this article, derivative estimations up to the third order (in root finding, some new initiatives) are applied in Taylor’s approximation of a nonlinear function / equation to achieve efficient iterative methods. Competent methods of higher orders for solving simple roots of nonlinear equations, which improve convergence of some basic existing methods, are investigated. We shall offer several examples for test of efficiency and convergence analyses, in C++. We also provide some examples of engineering applications of root finding. Graphical demonstrations are using matlab basic tools.
A new optimal root-finding iterative algorithm: local and semilocal analysis with polynomiography
Numerical Algorithms
In this work, a new optimal iterative algorithm is presented with fourth-order accuracy for root-finding of real functions. It uses only function as well as derivative evaluation. The algorithm is obtained as a combination of existing third-order methods by specifying a parameter involved. The algorithm is based on local and semilocal analysis and has been specifically designed to improve efficiency and accuracy. The proposed algorithm represents a significant improvement over existing iterative algorithms. In particular, it is tested on a range of polynomial functions and was found to produce accurate and efficient results, with improved performance over existing algorithms in terms of both speed and accuracy. The results demonstrate the effectiveness of the proposed algorithm and suggest that it has great potential for use in a wide range of applications in polynomiography and other areas of mathematical analysis.
Computers, Materials & Continua
In this article, we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear equations. Convergence analysis proved that the order of convergence of the family of derivative free simultaneous iterative method is nine. Our main aim is to check out the most regularly used simultaneous iterative methods for finding all roots of non-linear equations by studying their dynamical planes, numerical experiments and CPU time-methodology. Dynamical planes of iterative methods are drawn by using MATLAB for the comparison of global convergence properties of simultaneous iterative methods. Convergence behavior of the higher order simultaneous iterative methods are also illustrated by residual graph obtained from some numerical test examples. Numerical test examples, dynamical behavior and computational efficiency are provided to present the performance and dominant efficiency of the newly constructed derivative free family of simultaneous iterative method over existing higher order simultaneous methods in literature.
Root Finding by High Order Iterative Methods Based on Quadratures
Applied Mathematics and Computation
We discuss a recursive family of iterative methods for the numerical approximation of roots of nonlinear functions in one variable. These methods are based on Newton-Cotes closed quadrature rules. We prove that when a quadrature rule with n+1n+1n+1 nodes is used the resulting iterative method has convergence order at least n+2n+2n+2, starting with the case n=0n=0n=0 (which corresponds to the Newton's method).
On numerical schemes for determination of all roots simultaneously of non-linear equation
Mehran University Research Journal of Engineering and Technology, 2022
In this article, we first construct family of two-step optimal fourth order iterative methods for finding single root of non-linear equation. We then extend these methods for determining all the distinct as well as multiple roots of single variable non-linear equation simultaneously. Convergence analysis is presented for both the cases to show that the optimal order of convergence is 4 in case of single root finding method and 6 for simultaneous determination of all distinct as well as multiple roots of a non-linear equation. The computational cost, basins of attraction, computational efficiency, log of residual fall and numerical test functions validate that the newly constructed methods are more efficient as compared to the existing methods in the literature.
A novel cubically convergent iterative method for computing complex roots of nonlinear equations
Keywords: Root of continuous functions Taylor expansion Real and complex root Number of iterations a b s t r a c t A fast and simple iterative method with cubic convergent is proposed for the determination of the real and complex roots of any function F(x) = 0. The idea is based upon passing a defined function G(x) tangent to F(x) at an arbitrary starting point. Choosing G(x) in the form of x k or k x , where k is obtained for the best correlation with the function F(x), gives an added freedom, which in contrast to all existing methods, accelerates the convergence. Also, this new method can find complex roots just by a real initial guess. This is in contrast to many other methods like the famous Newton method that needs complex initial guesses for finding complex roots. The proposed method is compared to some new and famous methods like Newton method and a modern solver that is fsolve command in MATLAB. The results show the effectiveness and robustness of this new method as compared to other methods.
We construct a family of two-step optimal fourth order iterative methods for finding single root of non-linear equations. We generalize these methods to simultaneous iterative methods for determining all the distinct as well as multiple roots of single variable non-linear equations. Convergence analysis is present for both cases to show that the order of convergence is four in case of single root finding method and is twelve for simultaneous determination of all roots of non-linear equation. The computational cost, Basin of attraction, efficiency, log of residual and numerical test examples shows, the newly constructed methods are more efficient as compared to the existing methods in literature.