Solving Equations 1.1 Bisection Method 1.2 Fixed-Point Iteration 1.3 Limits of Accuracy 1.4 Newton's Method 1.5 Root-Finding without Derivatives Solving Equations (original) (raw)
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The study is aimed at comparing the rate of performance, viz-aviz, the rate of convergence of Bisection method, Newton-Raphson method and the Secant method of root-finding. The software, mathematica 9.0 was used to find the root of the function, f(x)=x-cosx on a close interval [0,1] using the Bisection method, the Newton's method and the Secant method and the result compared. It was observed that the Bisection method converges at the 52 second iteration while Newton and Secant methods converge to the exact root of 0.739085 with error 0.000000 at the 8 th and 6 th iteration respectively. It was then concluded that of the three methods considered, Secant method is the most effective scheme. This is in line with the result in our Ref. .
Improvements in the Bisection Method of finding roots of an equation
2014 IEEE International Advance Computing Conference (IACC), 2014
Bisection Method is one of the simplest methods in numerical analysis to find the roots of a non-linear equation. It is based on Intermediate Value Theorem. The algorithm proposed in this paper predicts the optimal interval in which the roots of the function may lie and then applies the bisection method to converge at the root within the tolerance range defined by the user. This algorithm also calculates another root of the equation, if that root lies just outside the range of the interval found.
Comparative Study of Bisection and Newton-Rhapson Methods of Root-Finding Problems
mekele
This paper presents two numerical techniques of root-finding problems of a non-linear equations with the assumption that a solution exists, the rate of convergence of Bisection method and Newton-Rhapson method of root-finding is also been discussed. The software package , MATLAB 7.6 was used to find the root of the function, f (x) = cosx − x * exp(x) on a close interval [0, 1] using the Bisection method and Newton's method the result was compared. It was observed that the Bisection method converges at the 14 th iteration while Newton methods converge to the exact root of 0.5718 with error 0.0000 at the 2 nd iteration respectively. It was then concluded that of the two methods considered, Newton's method is the most effective scheme. This is in line with the result in our Ref.[9].
International Journal of Latest Research in Engineering and Technology, 2019
Bisection Method for continuous functions and Newton-Raphson method for differentiable function are widely used for finding zeros of non-linear equations. Numerical techniques are explored when an analytic solution is not obvious. There is no single algorithm that works best for every function. We designed and implemented a new hybrid algorithm that is a dynamic blend of theBisectionand RegulaFalsialgorithms. The experimental results validate that the new algorithm outperforms Bisection,RegulaFalsi, Dekker's and Brent'salgorithms with respect to computational iterations. The new algorithm is guaranteed to find a root and requires fewer computational iterations. It is also observed that the new hybrid algorithm performs better that the Secant algorithm and the Newton-Raphson algorithm.The new algorithm guarantees the reliability of Bisection method and speed of Secant method. The theoretical and empirical evidence shows that the computational complexity of the new algorithm is considerably less than that of the classical algorithms.
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.
The n-th section method: A modification of Bisection
Malaysian Journal of Fundamental and Applied Sciences, 2017
Bisection method is the easiest method to find the root of a function. This method is based on the existence of a root on a specified interval. This interval is then halved or divided into two parts. The root is known to be laying in either one of these interval. The iterative sequence is continued until a desired stopping criterion is reached. In this research, a new modification of bisection method namely fourth section and sixth section methods are introduced. These methods are tested for several selected functions by using Maple software. The results are then analyzed based on the number of iterations and the CPU times. Based on the results, it is shown that when the interval increases, the CPU will also increase. However, the number of iterations is reduced significantly.
The bisection method is a bracketing method. This technique is also called the interval halving method because the interval is always divided in half as will be discussed in the coming slides. When applying the graphical technique, we have observed that changed sign on opposite sides of the root. In general, if is real and continuous in the interval from to and () and () have opposite signs, that is: ()) < 0 2 CE-301 Dr. Amin Abo-Monasar (UOHB)