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Topic: Bisection method

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	Bisection method - Wikipedia, the free encyclopedia
		In mathematics, the bisection method is a root-finding algorithm which works by repeatedly dividing an interval in half and then selecting the subinterval in which the root exists.
		The bisection algorithm is then applied to the sub-interval where the sign change occurs, meaning that the bisection algorithm is inherently recursive.
		The bisection method is less efficient than Newton's method but it is much less prone to odd behavior.
	en.wikipedia.org /wiki/Bisection_method (368 words)

	Nicholas Exner
		This is a detailed examination of four methods of discovering roots and the effects of different equations on the efficiency of each method and the amount of error produced by each method.
		An examination of the computer code for the false position method shows that it is very similar to the bisection method except that it uses the zero of the secant line to approach the root rather then finding the midpoint each time.
		The errors for all the methods are the same except for the bisection method.
	www.mste.uiuc.edu /activity/bisection/details.html (1638 words)

	Numerical Root Finding
		To improve the slow convergence of the bisection method, the secant method assumes that the function is approximately linear in the local region of interest and uses the zero-crossing of the line connecting the limits of the interval as the new reference point.
		However, since the secant method does not always bracket the root, the algorithm may not converge for functions that are not sufficiently smooth.
		The only difference between these two methods is that the secant method keeps the most recent two estimates, while the false position method retains the most recent estimate and the next recent one which has an opposite sign in the function value.
	www.efunda.com /math/num_rootfinding/num_rootfinding.cfm (482 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		So, the bisection method was not applicable for most intervals, but when it was applicable, it would always converge although in the most iterations and in the longest time out of the other root finding methods.
		Unlike the Bisection Method, the roots that it found were not constrained by its starting point, but the starting point tended to cause the method to find roots nearest to it.
		The Secant Method is similar to the Newton Method except that it uses an approximation of the derivative using two points on the function.
	www.cse.ohio-state.edu /~yates/lab1_541/writeup.txt (454 words)

	MTH 207 Lab Lesson 12 - The Bisection Method (Site not responding. Last check: 2007-11-04)
		Thus the error in the Bisection method decreases linearly.
		Rewrite the algorithm for the Bisection method to include a counter, which counts the number of iterations through the loop.
		The bisection method tends to be slow, needing a large number of iterations relative to other methods.
	www.sfu.ca /~rpyke/macm316/maple/less13.htm (697 words)

	Bisection Method for finding square roots
		This binary search method can be used not only to find the square-root of a positive number, but also for finding the zeros of any "continuous" function f.
		That is to scan the entire domain of the function by small increments and observe the steps in which a change of sign in the function f(x) occurs.
		The binary search method divides the interval of search by 2 and retains that half of the search interval in which the change of sign will occur.
	www.mathpath.org /Algor/squareroot/algor.square.root.binary.htm (751 words)

	The Method of Bisection (Site not responding. Last check: 2007-11-04)
		In the method of bisection, we approximate a solution of the equation f(x) = 0, where f(x) is a continuous function, by means of a successive reduction of intervals that contain the solution.
		The method of bisection will always produce an approximation of a root of a continuous function f(x) provided one can find values a and b, in the domain of f(x), such that f(a) < 0 and f(b) > 0.
		Because of this, the method of bisection is commonly utilized for the purpose of reducing the size of the interval on which a root of a given function is known to exist, and then an alternative method of root approximation is utilized to bring the approximation to within the desired amount of accuracy.
	www.mathcs.emory.edu /ccs/ccs215/roots/node2.html (875 words)

	Regula Falsi Method for finding square roots
		The methods differ only in how the next approximation is generated from the endpoints a and b.
		If you compare this table with the table for approximations to √5 given in the Bisection Method, for same starting values for L and H, you will find that the Regular Falsi method is faster.
		That is to say, the number of zeros in the error in the Bisection Method is guaranteed to increase by one after every four steps, for sufficiently large n.
	www.mathpath.org /Algor/squareroot/algor.square.root.regula.falsi.htm (1242 words)

	7.2.2 Newton's Method and the Secant Method (Site not responding. Last check: 2007-11-04)
		The bisection method is a very intuitive method for finding a root but there are other ways that are more efficient (find the root in fewer iterations).
		Implementation of Newton's method is left as an assignment for the student.
		The secant method is really the same method as Newton's method except the exact derivative is replaced with an approximate, ``numerical'' derivative.
	www.physics.nau.edu /~hart/matlab/node52.html (597 words)

	The Bisection Method
		The bisection method is one of the bracketing methods for finding roots of equations.
		Investigate the result of applying the bisection method over an interval where there is a discontinuity.
		Apply the bisection method for a function using an interval where there are distinct roots.
	math.fullerton.edu /mathews/n2003/BisectionMod.html (203 words)

	Bisection - Bisection Method Nonlinear Equations (Site not responding. Last check: 2007-11-04)
		Bisection method is the simplest method of bracketing the roots of a function and requires an initial interval which is guaranteed to contain a root.
		Bisection is the division of a given curve, figure, or interval into two equal A simple bisection procedure for iteratively converging on a solution
		bisection method In mathematics, the bisection method is a root-finding algorithm which works by repeatedly dividing an interval in half and then.
	parkinter.com /?q=bisection (547 words)

	Solver Version 3
		Newton's method uses calculus in the computation of slopes of tangent lines.
		The bisection method does not use any calculus and usually takes longer to converge.
		In the sheet Bisection Method the initial left-hand endpoint, L1, is specified as 3 in the cell for "lower x".
	faculty.babson.edu /aieta/exclcmpn/AppndxD/AppndixD.htm (647 words)

	Finding Roots by "Closed" Methods
		These methods are guaranteed to find a root within the interval, as long as the function is well-behaved.
		The two closed methods discussed here also require only that you be able to evaluate the function at some point; they do not require that you also be able to evaluate the function's derivative.
		The method assumes that the function changes linearly from one end to the other, and calculates the value of x_new at which that linear approximation crosses zero.
	spiff.rit.edu /classes/phys317/lectures/closed_root/closed_root.html (955 words)

	Bisection Method E-book Nonlinear Equations
		Bisection Method, Numerical Solution of Nonlinear Equations, Mathcad, Maple, Mathematica, Matlab, Simulations.
		Since the method is based on finding the root between two points, the method falls under the category of bracketing methods.
		Table 1: Root of f(x)=0 as function of number of iterations for bisection method.
	numericalmethods.eng.usf.edu /ebooks/bisection_03nle_ebook.htm (882 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-04)
		The great thing about this method is that it works for everything, at least everything normal that you're likely to run into at this point in your mathematical education.
		The usual method is to first plot a function to get an idea about where the roots are.
		Where "bisection" comes in is that the best way to do this is to pick the midpoint of the interval you know the root to lie in, just as the contestant picked midway between the price range she knew.
	forum.swarthmore.edu /dr.math/problems/jcg8.1.98.html (545 words)

	Bisection Method Tutorial
		The bisection method is discussed in Chapter 9 as a way to solve equations in one unknown that cannot be solved symbolically.
		To solve this equation using the bisection method, we first manipulate it algebraically so that one side is zero.
		The bisection method locates such a root by repeatedly narrowing the distance between the two guesses.
	www.cs.utah.edu /~zachary/isp/applets/Root/Bisection.html (718 words)

	The Bisection Method (Site not responding. Last check: 2007-11-04)
		The bisection method is developed with the support of the Intermediate-Value Theorem, 1.3.
		At each step the interval in which there is guaranteed to be a root of the equation is halved (bisected), and the method terminates as soon as the width of the interval containing the root is less than some error tolerance
		Since the bisection method keeps a bounded interval where there is at least one root at each step, it falls in the category of bracketing methods.
	www.mathcs.emory.edu /ccs/ccs315/ccs315/node14.html (248 words)

	LAB #4: Fast & Dangerous Root Finding
		For the bisection method, the most natural test is on the size of the change-of-sign interval.
		Newton's method is using a (linear) model of the unknown function, but its "evidence" is now all gathered at the latest point.
		Muller's method determines the quadratic polynomial that passes through these three points, and then solves for the roots of that polynomial, and chooses one of them to add as its latest point, while discarding the oldest point.
	orion.math.iastate.edu /burkardt/math2070/lab_04.html (1903 words)

	Proof for Brent's Method
		The regula falsi method usually converge faster than the bisection method bisection.
		However, examples can be found when the bisection method converges faster.
		The method uses a combination of three methods: bisection, regula falsi, and inverse quadratic interpolation.
	math.fullerton.edu /mathews/n2003/BrentMethodProof.html (365 words)

	MATH-305, Prof. K. TeBeest (Site not responding. Last check: 2007-11-04)
		If we use the Bisection method to approximate a zero of a function on the interval [-2,3], what is the error bound after 12 iterations?
		use the Bisection method's error bound formula to determine the number of iterations required to guarantee that we would approximate a zero with an accuracy of at least 7 decimal places if we were to start on the interval [2,7].
		Write a computer program that uses the Bisection method (get the FORTRAN code here or the generic algorithm here) to approximate a zero of a function f(x) on an interval [a,b].
	www.kettering.edu /~ktebeest/math305/sec1_1.html (215 words)

	Bisection Methods:
		Hence we choose the subinterval [1/3,2/3] and bisect it further.
		This is generally true of numerical methods for solving nonlinear equations.
		Thus the choice of starting interval is important to the success of the bisection method.
	web.mit.edu /10.001/Web/Course_Notes/NLAE/node2.html (362 words)

	Numerical Methods Lecture Notes: roots
		The guaranteed convergence of the bisection method does not require such safety checks which, combined with its extreme simplicity, is one of the reasons for its widespread use despite being relatively slow to converge.
		This method is similar to the bisection method in that it requires two initial guesses to bracket the root.
		A graphical representation of the method working is shown in figure 5 and failure to converge in figure 6.
	www.damtp.cam.ac.uk /user/fdl/people/sd103/lectures/nummeth98/roots.htm (1243 words)

	Jesse Enloe \| Papers \| Numerical Analysis \| Iterative Root Finding Techniques (Site not responding. Last check: 2007-11-04)
		The Bisection Method was found be the most expensive of all three (meaning it required the greatest number of iterations), and converged the slowest - however, if a zero did exist in the given interval, the method would always converge to that root.
		The Newton Method was found to be the most accurate in that the average of all F(root) it determined was closest to zero.
		It was determined that the Secant Method converged to a solution with the least expense, recquiring a less number of iterations than either the Bisection Method or the Newton-Raphson Method.
	www.coaps.fsu.edu /~enloe/Papers/Numanal/Numanal3 (832 words)

	Bisection Method In Matlab (Site not responding. Last check: 2007-11-04)
		In layman's terms, a method is a series of steps taken to accomplish an objective.
		In computer science, a method is another name for an action, algorithm, function, or procedure ; more specifically, in object-oriented programming, it is an implementation of code responding to certainmessages.
		Method acting is a style of acting in which the actor attempts to replicate the conditions under which the character operates.
	www.vermontreview.com /edge/48719-bisection%20method%20in%20matlab.html (144 words)

	Equation Solving
		Note that, with false position, we are guaranteed that our range always spans the root, and convergence is assured, although the method is generally a little slower than the secant method.
		One can construct situations where the secant method performs far worse than bisection but, as a rule of thumb, it can be shown that, once we are close to a root, the secant method more than doubles the number of digits of accuracy of the answer every two iterations.
		In this method, instead of doing linear interpolation between two points known to straddle the root, as in the secant method, we use the value of the function
	www.physics.drexel.edu /courses/Comp_Phys/BV/root.html (699 words)

	CE 311K Lab 6 (Site not responding. Last check: 2007-11-04)
		In this laboratory, we will explore a method that we have considered for solving nonlinear equations, the bisection method.
		The essence of the bisection method lies in the fact that the sign of a function f(x) changes on opposite sides of a root.
		The bisection method is based on the fact that when an interval [a,c] contains a root, the sign of the function at the two ends (f(a) and f(c)) are opposite each other, namely
	www.ce.utexas.edu /prof/mckinney/ce311K/Lab/Lab9/Lab9.html (651 words)

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