Eich-Soellner, Edda · Acervo Digital da Universidade Estadual Paulista (Universidade Estadual Paulista) · 2016
This Demonstration shows the steps of the bisection root-finding method for a set of functions. You can choose the initial interval by dragging the vertical, dashed lines. Each iteration step halves the current interval into two subintervals; the next interval in the sequence is the subinterval with a sign change for the function (indicated by the red horizontal lines). The method always converges to a root of f if f is continuous and f(a) and f(b) have opposite sign
Cited by 0
Eich-Soellner, Edda · Acervo Digital da Universidade Estadual Paulista (Universidade Estadual Paulista) · 2011
This Demonstration shows the steps of the bisection root-finding method for a set of functions. You can choose the initial interval by dragging the vertical, dashed lines. Each iteration step halves the current interval into two subintervals; the next interval in the sequence is the subinterval with a sign change for the function (indicated by the red horizontal lines). The method always converges to a root of f if f is continuous and f(a) and f(b) have opposite sign
Cited by 0
Shams El-Din El-Fouly, Siddharth Misra · SHAREOK (University of Oklahoma) · 2020
In this lesson you'll learn about the different types of Root of Equations techniques, the bisection method and how to develop a VBA code to implement this technique to any desired approximate error.
Cited by 0Open access
Lavkush Pandey, Trilokinath · THE SCIENTIFIC TEMPER · 2022
Fourth roots of the natural numbers from 1 to 30 have been calculated by Bisection method inthe interval [0, 3] using stopping tolerance 0f 0.00001. Calculated roots have been comparedwith the actual values of roots to obtain error and percentage error in the calculated roots.Numerical rate of convergence has also been calculated in the determination of each fourthroot. The highest numerical rate of convergence of Bisection method has been observed in thecalculation of fourth root of 2 and is equal to 1.754385964912. The lowest numerical rate ofconvergence of Bisection method has been observed in the calculation of fourth roots of 1, 3,4-8, 10, 12 and is equal to 1.333333333333. Average error, average percentage error and averagenumerical rate of convergence of Bisection method have been found to be 0.000000062635,0.000003048055 and 1.458082183940 respectively
Cited by 0Open access
Gutiérrez Gallardo, Claudio, Gutiérrez, Flavio, Rivara Zúñiga, María Cecilia · Universidad de Chile · 2009
The bisection method is the consecutive bisection of a triangle by the median of the longest side. In this paper we prove a \nsubexponential asymptotic upper bound for the number of similarity classes of triangles generated on a mesh obtained by iterative \nbisection, which previously was known only to be finite. The relevant parameter is \n / , where \n is the biggest and is the smallest \nangle of the triangle.We get this result by introducing a taxonomy of triangles that precisely captures the behaviour of the bisection \nmethod.We also prove that the number of directions on the plane given by the sides of the triangles generated is finite. Additionally, \nwe give purely geometrical and intuitive proofs of classical results for the bisection method.
Cited by 0
Claudio Gutiérrez, Flavio Gutierrez, Maŕıa-Cecilia Rivara · Theoretical Computer Science · 2007
Cited by 26
Shefali Kulkarni-thaker · 2013
Consider the following unconstrained optimization problem min f(x) x ∈ R Any optimization algorithm starts by an initial point x0 and performs series of iterations to reach the optimal point x ∗. At any k th iteration the next point is given by the sum of old point and the direction in which to search of a next point multiplied by how far to go in that direction. Thus xk+1 = xk + αkdk here dk is a search direction and αk is a positive scalar determining how far to go in that direction. It is called as the step length. Additionally, the new point must be such that the function value (the function which we are optimizing) at that point should be less than or equal to the previous point. This is quite obvious because if we are moving in a different direction where the function value is increasing we are not really moving towards the minimum. Thus, f(xk+1) ≤ f(xk) f(xk + αkdk) ≤ f(xk) The general optimization algorithm begins with an initial point, finds a descent search direction, determines the step length and checks the termination criteria. So, when we are trying to find the step length, we already know that the direction in which we are going is descent. So, ideally we want to go far enough so that the function reaches its minimum. Thus, given the previous point and a descent search direction, we are trying to find a scalar step length such that the value of the function is minimum in that direction. In its mathematical form we may write, min f(x + αd) α≥0 Since, x and d are known, this problem reduces to a univariate, 1−D, minimization problem.Assuming that f is smooth and continuous, we find its optimum where its first-derivative is zero, i.e. ∇f(x + αd) = 0. All we are doing is trying to find zero of a function (i.e. the point where the curves intersects the x−axis). We will visit some known and some newer zero-finding (or root-finding) techniques. When, the dimensionality of the problem or the degree of equation increases finding an exact zero is difficult. Usually, we are looking for an interval 0 ∈ [a, b] so that |a − b | < ɛ, where ɛ is an acceptable tolerance. 1 Bisection Method In bisection method we reduce begin with an interval so that 0 ∈ [a, b] and divide the interval in two halves,i.e. [a, a+b a+b] and [ , b]. A next search interval is chosen by comparing and finding which one
Cited by 0
Graham R. Wood · Mathematical Programming · 1992
Cited by 94
Max Demenkov · IFAC Proceedings Volumes · 2007
Cited by 2
Nasr Al, Din Ide, Nasr Al Din Ide et al. · International Journal of Scientific and Innovative Mathematical Research · 2019
Finding the roots of nonlinear algebraic equations is an important problem in science and engineering, later many methods have been developed by using Taylor's series, decomposition, homotopy, for solving nonlinear equations . Fuzzy numbers are foundation of fuzzy sets and fuzzy mathematics that extends the domain of numbers from real numbers to fuzzy numbers. Several researchers investigated a number of methods of numerical analysis by using Fuzzy theory. Recently, various methods have been developed for solving linear programming problems with using fuzzy number. Many research works were done on fuzzy numbers and on its applications in various field Toralima Bora and G.C. Hazarika [26], Treat the fuzzification of Newton Raphson method to find the solution of cubic equation in the form of triangular numbers along with the membership function. In this work the fuzzification of Bisection method to give the solution of nonlinear equations. Results have been obtained in the form of triangular numbers with the membership function also. These methods based on the method given by Toralima Bora and G.C. Hazarika [26], 2017. We verified the work by an example and numerical results obtained to clarify the present method.
Cited by 1Open access
K S Varun · 2025
The Otsu thresholding algorithm represents a fundamental technique in image segmentation, yet its computational efficiency is severely limited by exhaustive search requirements across all possible threshold values. This work presents an optimized implementation that leverages the bisection method to exploit the unimodal characteristics of the between-class variance function. Our approach reduces the computational complexity from O(L) to O(log L) evaluations while preserving segmentation accuracy. Experimental validation on 48 standard test images demonstrates a 91.63% reduction in variance computations and 97.21% reduction in algorithmic iterations compared to conventional exhaustive search. The bisection method achieves exact threshold matches in 66.67% of test cases, with 95.83% exhibiting deviations within 5 gray levels. The algorithm maintains universal convergence within theoretical logarithmic bounds while providing deterministic performance guarantees suitable for real-time applications. This optimization addresses critical computational bottlenecks in large-scale image processing systems without compromising the theoretical foundations or segmentation quality of the original Otsu method.
Cited by 0Open access
Prassan Sharma · Pranjana The Journal of Management Awareness · 2023
The essential way of getting a root is the bisection method. When iterations are applied, the interval length gets hafted. Therefore, this Method guarantees the convergence of a root to "g" if the given function "g" is continuous at a [a,b] and the signs of g(a) and g(b) are opposite.This Research paper explains the role of the binary search method in computer science. We presented a modern strategy, which may be a mixture of binary search and other strategies, to demonstrate that with the assistance of the binary search strategy, we can create modern strategies. It is shown that researchers and engineers faced problems in finding the roots of equations by using a basic method, which is a binary search method, but it is equivalently leaden. We can present this recent approach to solve such issues and to enhance the convergence.
Cited by 1
Journal of Mathematical and Computational Science · 2021
The purpose of this research propose is to hybrid algorithm to Newton-Raphson method and Bisection method to compute roots of nonlinear equations. Numerical experiments for various tests nonlinear equations confirm performance for the bracketing method or open method to be compared with.
Cited by 1Open access
BILLIANTO, ARDY · Unika Repositor (Unika) · 2015
High rank equation is a function to search for the roots of an equation \nresults . In the high rank of the equation can be solved by some means or \nmethods . The method can be used to solve the equations of high rank is a \nnumerical method , graphical method , bisection method , secant method and \nNewton-Raphson method . \nThe method to be used is the bisection method. Bisection is a concept to \ncalculate the result of the roots of equation of High Rank. Bisection method is \nalso called the method divided by two. So how to find the end result is to use \nthe middle value or the result of the division of two values. This method is \nused because it is easier and quicker to understand but longer in the \ncalculations . \nBisection method implementation is done by using the c language. The end \nresult of this process is to find the value bisection x average to find a point x \nis close to zero
Cited by 0Open access
Younbae Jun, Junho Jeon, M Basto et al. · International Journal of Scientific and Innovative Mathematical Research · 2019
This method is one of the numerical methods that is an extension of bisection method to solve nonlinear equations. The paper deals with the convergence and number of iterations with visual graph.
Cited by 3Open access
American Mathematical Monthly · 1990
Cited by 0
Arthur S. Finbow · American Mathematical Monthly · 1985
Cited by 2
Ludovick Bouthat, Philippe-André Luneau, Philippe Petitclerc · ArXiv.org · 2026
We propose a generalized version of the bisection method where the cutting point between the two subintervals is chosen at random following an arbitrary distribution. We compute expected convergence rates with respect to any arbitrary a priori distribution for the position of the root in the initial interval and proved that it depends only on the the expectation $\mathbb{E}[c(1-c)]$ of the cut $c$. We also provide a generalization of the method for $K$ random cuts and study its convergence properties. Most probabilistic derivations are kept fairly simple for the ease of understanding of a larger audience. Our theoretical results are then validated numerically using statistical simulation.
Cited by 0Open access
Ludovick Bouthat, Philippe-André Luneau, Philippe Petitclerc · arXiv (Cornell University) · 2026
We propose a generalized version of the bisection method where the cutting point between the two subintervals is chosen at random following an arbitrary distribution. We compute expected convergence rates with respect to any arbitrary a priori distribution for the position of the root in the initial interval and proved that it depends only on the the expectation $\mathbb{E}[c(1-c)]$ of the cut $c$. We also provide a generalization of the method for $K$ random cuts and study its convergence properties. Most probabilistic derivations are kept fairly simple for the ease of understanding of a larger audience. Our theoretical results are then validated numerically using statistical simulation.
Cited by 0Open access
K. Sikorski, G. M. Trojan · Numerische Mathematik · 1990
Cited by 11