scholarly journals A Comparative Study among New Hybrid Root Finding Algorithms and Traditional Methods

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1306
Author(s):  
Elsayed Badr ◽  
Sultan Almotairi ◽  
Abdallah El Ghamry
Keyword(s):  
Comparative Study ◽  
Numerical Results ◽  
Traditional Methods ◽  
Running Time ◽  
Root Finding ◽  
Average Running Time ◽  
False Position ◽  
Newton Raphson ◽  
Number Of Iterations

In this paper, we propose a novel blended algorithm that has the advantages of the trisection method and the false position method. Numerical results indicate that the proposed algorithm outperforms the secant, the trisection, the Newton–Raphson, the bisection and the regula falsi methods, as well as the hybrid of the last two methods proposed by Sabharwal, with regard to the number of iterations and the average running time.

scholarly journals An Iterative Hybrid Algorithm for Roots of Non-Linear Equations

Eng ◽  
2021 ◽  
Vol 2 (1) ◽  
pp. 80-98
Author(s):  
Chaman Lal Sabharwal
Keyword(s):  
Hybrid Algorithm ◽  
Analytic Solution ◽  
Linear Equations ◽  
Root Finding ◽  
Engineering Sciences ◽  
Non Linear ◽  
False Position ◽  
Newton Raphson ◽  
And Performance ◽  
Non Linear Equations

Finding the roots of non-linear and transcendental equations is an important problem in engineering sciences. In general, such problems do not have an analytic solution; the researchers resort to numerical techniques for exploring. We design and implement a three-way hybrid algorithm that is a blend of the Newton–Raphson algorithm and a two-way blended algorithm (blend of two methods, Bisection and False Position). The hybrid algorithm is a new single pass iterative approach. The method takes advantage of the best in three algorithms in each iteration to estimate an approximate value closer to the root. We show that the new algorithm outperforms the Bisection, Regula Falsi, Newton–Raphson, quadrature based, undetermined coefficients based, and decomposition-based algorithms. The new hybrid root finding algorithm is guaranteed to converge. The experimental results and empirical evidence show that the complexity of the hybrid algorithm is far less than that of other algorithms. Several functions cited in the literature are used as benchmarks to compare and confirm the simplicity, efficiency, and performance of the proposed method.

scholarly journals Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation

Processes ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 130 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Iterative Procedure ◽  
Accurate Solution ◽  
Final Solution ◽  
Worst Case ◽  
Flow Friction ◽  
One Step ◽  
Newton Raphson ◽  
Number Of Iterations

The Colebrook equation is implicitly given in respect to the unknown flow friction factor λ; λ = ζ ( R e , ε * , λ ) which cannot be expressed explicitly in exact way without simplifications and use of approximate calculus. A common approach to solve it is through the Newton–Raphson iterative procedure or through the fixed-point iterative procedure. Both require in some cases, up to seven iterations. On the other hand, numerous more powerful iterative methods such as three- or two-point methods, etc. are available. The purpose is to choose optimal iterative method in order to solve the implicit Colebrook equation for flow friction accurately using the least possible number of iterations. The methods are thoroughly tested and those which require the least possible number of iterations to reach the accurate solution are identified. The most powerful three-point methods require, in the worst case, only two iterations to reach the final solution. The recommended representatives are Sharma–Guha–Gupta, Sharma–Sharma, Sharma–Arora, Džunić–Petković–Petković; Bi–Ren–Wu, Chun–Neta based on Kung–Traub, Neta, and the Jain method based on the Steffensen scheme. The recommended iterative methods can reach the final accurate solution with the least possible number of iterations. The approach is hybrid between the iterative procedure and one-step explicit approximations and can be used in engineering design for initial rough, but also for final fine calculations.

scholarly journals Research on Intelligent Solution of Service Industry Supply Chain Network Optimization Based on Genetic Algorithm

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Yixin Zhou ◽  
Zhen Guo
Keyword(s):  
Genetic Algorithm ◽  
Supply Chain ◽  
Network Optimization ◽  
Service Industry ◽  
User Behavior ◽  
Service Level ◽  
Supply Chain Network ◽  
Supply Network ◽  
Running Time ◽  
Average Running Time

With the advent of the era of big data (BD), people’’s living standards and lifestyle have been greatly changed, and people’s requirements for the service level of the service industry are becoming higher and higher. The personalized needs of customers and private customization have become the hot issues of current research. The service industry is the core enterprise of the service industry. Optimizing the service industry supply network and reasonably allocating the tasks are the focus of the research at home and abroad. Under the background of BD, this paper takes the optimization of service industry supply network as the research object and studies the task allocation optimization of service industry supply network based on the analysis of customers’ personalized demand and user behavior. This paper optimizes the supply chain network of service industry based on genetic algorithm (GA), designs genetic operator, effectively avoids the premature of the algorithm, and improves the operation efficiency of the algorithm. The experimental results show that when m = 8 and n = 40, the average running time of the improved GA is 54.1 s. The network optimization running time of the algorithm used in this paper is very fast, and the stability is also higher.

Comparative Study of Iterative Methods for Solving Non-Linear Equations

2021 ◽  
Vol 23 (07) ◽  
pp. 858-866
Author(s):  
Gauri Thakur ◽  
◽  
J.K. Saini ◽  
Keyword(s):  
Numerical Analysis ◽  
Iterative Methods ◽  
Linear Equations ◽  
Bisection Method ◽  
Practical Applications ◽  
Non Linear ◽  
False Position ◽  
Newton Raphson ◽  
Raphson Method ◽  
Non Linear Equations

In numerical analysis, methods for finding roots play a pivotal role in the field of many real and practical applications. The efficiency of numerical methods depends upon the convergence rate (how fast the particular method converges). The objective of this study is to compare the Bisection method, Newton-Raphson method, and False Position Method with their limitations and also analyze them to know which of them is more preferred. Limitations of these methods have allowed presenting the latest research in the area of iterative processes for solving non-linear equations. This paper analyzes the field of iterative methods which are developed in recent years with their future scope.

scholarly journals Visual Analysis of the Newton’s Method with Fractional Order Derivatives

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1143 ◽  
Author(s):  
Krzysztof Gdawiec ◽  
Wiesław Kotarski ◽  
Agnieszka Lisowska
Keyword(s):  
Visual Analysis ◽  
Dynamic Properties ◽  
Speed Of Convergence ◽  
Polynomial Zeros ◽  
Polynomial Roots ◽  
Root Finding ◽  
Root Finding Method ◽  
Finding Method ◽  
Number Of Iterations ◽  
Basins Of Attractions

The aim of this paper is to investigate experimentally and to present visually the dynamics of the processes in which in the standard Newton’s root-finding method the classic derivative is replaced by the fractional Riemann–Liouville or Caputo derivatives. These processes applied to polynomials on the complex plane produce images showing basins of attractions for polynomial zeros or images representing the number of iterations required to obtain polynomial roots. These latter images were called by Kalantari as polynomiographs. We use both: the colouring by roots to present basins of attractions, and the colouring by iterations that reveal the speed of convergence and dynamic properties of processes visualised by polynomiographs.

Running-time Analysis of Ant System Algorithms with Upper-bound Comparison

2017 ◽  
Vol 8 (4) ◽  
pp. 1-17
Author(s):  
Han Huang ◽  
Hongyue Wu ◽  
Yushan Zhang ◽  
Zhiyong Lin ◽  
Zhifeng Hao
Keyword(s):  
Optimal Solution ◽  
Numerical Results ◽  
Ant System ◽  
Time Analysis ◽  
Traveling Salesman Problems ◽  
Running Time ◽  
Cycle System ◽  
Running Time Analysis ◽  
Global Optimal

Running-time analysis of ant colony optimization (ACO) is crucial for understanding the power of the algorithm in computation. This paper conducts a running-time analysis of ant system algorithms (AS) as a kind of ACO for traveling salesman problems (TSP). The authors model the AS algorithm as an absorbing Markov chain through jointly representing the best-so-far solutions and pheromone matrix as a discrete stochastic status per iteration. The running-time of AS can be evaluated by the expected first-hitting time (FHT), the least number of iterations needed to attain the global optimal solution on average. The authors derive upper bounds of the expected FHT of two classical AS algorithms (i.e., ant quantity system and ant-cycle system) for TSP. They further take regular-polygon TSP (RTSP) as a case study and obtain numerical results by calculating six RTSP instances. The RTSP is a special but real-world TSP where the constraint of triangle inequality is stringently imposed. The numerical results derived from the comparison of the running time of the two AS algorithms verify our theoretical findings.

scholarly journals Blended Root Finding Algorithm Outperforms Bisection and Regula Falsi Algorithms

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1118
Author(s):  
Sabharwal
Keyword(s):  
Computational Complexity ◽  
Empirical Evidence ◽  
Analytic Solution ◽  
Fundamental Problem ◽  
Numerical Techniques ◽  
Physical Sciences ◽  
Root Finding ◽  
Numerical Computing ◽  
Newton Raphson ◽  
Raphson Algorithm

Finding the roots of an equation is a fundamental problem in various fields, including numerical computing, social and physical sciences. Numerical techniques are used when an analytic solution is not available. There is not a single algorithm that works best for every function. We designed and implemented a new algorithm that is a dynamic blend of the bisection and regula falsi algorithms. The implementation results validate that the new algorithm outperforms both bisection and regula falsi algorithms. It is also observed that the new algorithm outperforms the secant algorithm and the Newton–Raphson algorithm because the new algorithm requires fewer computational iterations and is guaranteed to find a root. The theoretical and empirical evidence shows that the average computational complexity of the new algorithm is considerably less than that of the classical algorithms.

Sign in / Sign up

Export Citation Format

Share Document