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Topic: Secant method

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	Secant method - Wikipedia, the free encyclopedia
		In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f.
		The secant method does not require that the root remain bracketed like the bisection method does.
		The false position method, which is based on the secant method, avoids this problem by checking the brackets at each step.
	en.wikipedia.org /wiki/Secant_method (480 words)

	Encyclopedia article: Secant method (Site not responding. Last check: 2007-10-08)
		Note that this line is a secant (Ratio of the hypotenuse to the adjacent side of a right-angled triangle) or chord of the graph of the function f.
		The secant method does not require that the root remain bracketed like the bisection method (additional info and facts about bisection method) does.
		The false position method (additional info and facts about false position method), which is based on the secant method, avoids this problem by checking the brackets at each step.
	www.absoluteastronomy.com /encyclopedia/s/se/secant_method.htm (495 words)

	Secant Method (Site not responding. Last check: 2007-10-08)
		Such a method is the secant method which closely resembles the regula falsi method in using a linear approximation to f(x) at each iteration, but with the bracketing aspect dropped.
		The secant method is closely related to the Newton-Raphson method, and this relationship is clearly related to the fact that the quantity
		The regula falsi method is sure to find the zero, since it keeps it bracketed, while the secant method can sometimes fail to find a zero that does exist.
	pathfinder.scar.utoronto.ca /~dyer/csca57/book_P/node36.html (642 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)

	MCS 471 Lecture Two (Site not responding. Last check: 2007-10-08)
		The focus of this lecture was on the secant method.
		Thus the secant method is faster than the bisection method, but note that it is quite easy (as we did in class) to construct examples where the secant method fails.
		We stated the theorem and outlined the proof of "the convergence of the secant method is superlinear".
	www.math.uic.edu /~jan/mcs471/Lec5/lec5.html (369 words)

	Nicholas Exner (Site not responding. Last check: 2007-10-08)
		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.
		Unlike the other two methods, f(a) and f(b) of the two initial values that we pick to start with do not have be on opposites sides of the root and have a sign change.
	www.mste.uiuc.edu /activity/bisection/details.html (1638 words)

	Numerical Searching Method - Secant Method
		Secant method, unlike the Newton-Ralphson method, does not require the differentiation of the equation in question.
		The example below demostrates the application using Secant method to solve for 2 equations (both equations are set to zero) individually.
		The method returns -1.59488 as the value of the root for the first equation.
	www.anthony-vba.kefra.com /vba/vban2.htm (309 words)

	Methods for Solving Nonlinear Equations
		In one dimension, the idea of the secant method is to use the slope of the line between two consecutive search points to compute the step instead of the derivative at the latest point.
		Brent's method [Br02] is effectively a safeguarded secant method that always keeps a point where the function is positive and one where it is negative so that the root is always bracketed.
		The method gives up and issues a message when the root is bracketed very closely, but it is not able to find a value of the function, which is zero.
	documents.wolfram.com /v5/Built-inFunctions/AdvancedDocumentation/Optimization/UnconstrainedOptimization/1.4.html (1261 words)

	PHY498CQM Lecture Notes 2 - Root finding (Site not responding. Last check: 2007-10-08)
		This method has "linear" convergence, that is, the log of the error decreases linearly with the number of iterations.
		The secant method uses the previous point in the sequence for the approximation of the derivative.
		The 'standard' (Brent's method) for calculating roots of a nonlinear equation without using derivatives is a hybrid method, combining root bracketing, bisection, and inverse quaderature interpolation (instead of the linear interpolation used in our example).
	www.physics.uiuc.edu /research/ElectronicStructure/498CQM/lnotes/lec2.html (474 words)

	The Secant Method for square roots (Site not responding. Last check: 2007-10-08)
		The Secant Method is super-linear in the speed of convergence.
		Since the Secant Method is super-linear in speed of convergence, why use the Bisection Method or the Regula Falsi Method.
		This is not always the case with the Secant method.
	www.mathpath.org /Algor/algor.square.root.secant.htm (427 words)

	Secant Method
		The secant method is an iterative procedure to estimate a root of an equation f(x) = 0 where the user gives two initial estimates and a tolerance.
		The method returns an err of 0 if the iteration was successful, -1 if the number of iterations exceed the maximum allowable number of iterations as specified by the user, and -2 if an attempt was made to divide by zero, which is indicative of a local minimum or a local maximum.
		The secant method is // // similar to the Newton-Raphson method in which the derivative is // // replaced by a numerical estimate of the derviative.
	mymathlib.webtrellis.net /roots/secant_method.html (767 words)

	Math Library - Source Code
		The most commonly used methods for finding a root of a nonlinear equation are the bisection method, secant methods, Newton-Raphson, and certain variants Mueller's method, van Wijngaarden,Dekker,Brent method.
		In the regula falsi method the secant is chosen from points on the curve corresponding to the most recent estimates which bracket a root.
		The Newton-Raphson method is similar to the secant method only instead of estimating a root by the intersection of the x-axis with a secant throught two points on the curve, the estimate is obtained from the intersection of the x-axis and the tangent to the curve at a point.
	mymathlib.webtrellis.net /roots.html (1061 words)

	Secant - Wikipedia, the free encyclopedia
		Secant, a trigonometric function, given by sec(x) = 1/cos(x).
		In numerical analysis, the secant method is a root-finding algorithm.
		This is a disambiguation page: a list of articles associated with the same title.
	en.wikipedia.org /wiki/Secant (90 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		The behavior of the secant method with changes in initial guesses is charted in the.jpg named secant_analysis.jpg This graphs the number of iterations required to reach solution (at 10^-6 error level) as a function of first and second guesses.
		The secant method is similarly limited by runaway behavior.
		Similarly, the secant method finds different zeroes depending upon the starting guesses; the shape of the curve doesn't allow the escape that tanh allows these algorithms.
	www.technion.ac.il /~rutman/comphy/targil3.1.description (408 words)

	GNU Scientific Library -- Reference Manual: Root Finding Algorithms using Derivatives
		The secant method is a simplified version of Newton's method does not require the computation of the derivative on every step.
		Asymptotically the secant method is faster than Newton's method whenever the cost of evaluating the derivative is more than 0.44 times the cost of evaluating the function itself.
		As with all methods of computing a numerical derivative the estimate can suffer from cancellation errors if the separation of the points becomes too small.
	linux.duke.edu /~mstenner/free-docs/gsl-ref-1.0/gsl-ref_407.html (444 words)

	7.2.2 Newton's Method and the Secant Method (Site not responding. Last check: 2007-10-08)
		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 (585 words)

	WebCab Functions for Delphi v2.0
		Secant Method - Generally this procedure converges and is much faster than the interval bisection method.
		Method of Regula Falsi - This procedure uses a slight alteration on the secant method to ensure convergence.
		The procedure is generally faster than the interval bisection method and slightly slower than the secant method.
	www.webcabcomponents.com /delphi/components/functions/index.shtml (662 words)

	[No title]
		Secant method The secant method also starts with two approximate solutions, x_0 and x_1.
		The line that passes through the points (x_0, f(x_0)) and (x_1, f(x_1)) is a secant of the curve defined by f.
		The secant method involves solving for x_{n+1} based on the secant defined by (x_n, f(x_n)) and (x_{n-1}, f(x_{n-1})).
	www.dgp.toronto.edu /people/JamesStewart/270/9697f/hwk1/hwk1.txt (882 words)

	LAB #4: Fast & Dangerous Root Finding
		The secant method retains the idea of using a linear model of the function.
		One explanation for the speedup of the secant method is the fact that it throws away old data.
		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)

	The Secant Method
		One drawback associated with Newton's method is the requirement that there be some mechanism to compute the derivative of the function.
		That is, the secant method may be viewed as a modification of Newton's method, where the modification takes the form of the approximation
		In other words, the secant method as described below is not a bracketing method, although variations on the bracketing requirements leads to alternate bracketing methods such as the ``Regula-Falsi'' (false-position) method.
	www.mathcs.emory.edu /ccs/ccs315/ccs315/node18.html (290 words)

	Brent's Method (Site not responding. Last check: 2007-10-08)
		We start by reviewing the secant method and then extend it to the inverse quadratic interpolation method.
		Mueller's proposed a method using successive bisection and inverse quadratic interpolation which was extended by Brent's and others.
		The method uses a combination of three methods: bisection, regula falsi, and inverse quadratic interpolation.
	math.fullerton.edu /mathews/n2003/BrentMethodMod.html (477 words)

	AMTH247 Lecture 15 Nonlinear Equations II
		One important property of the bisection algorithm, not shared by most other algorithms, is that it always converges provided the function is continuous and we start from interval on which the function has opposite signs at the two endpoints.
		The secant method is similar to Newton's method except that we approximate the function by a secant rather than by the tangent.
		A simple way to combine the bisection method with other more rapidly convergent algorithms is to start with an interval known to contain the solution and apply the faster method as long as it produces results in the interval.
	turing.une.edu.au /~amth247/Lectures_2003/Lecture_15/lecture (916 words)

	False position method
		In numerical analysis, the false position method or regula falsi method is a root-finding algorithm that combines features from the bisection method and the secant method.
		At some point, the root of the secant line may be a good enough approximation of the root of the function f.
		It was designed to solve the same problem as solved by the Newton's method and secant method code: to find the positive number x where cos(x) = x
	www.brainyencyclopedia.com /encyclopedia/f/fa/false_position_method.html (315 words)

	Jesse Enloe \| Papers \| Numerical Analysis \| Iterative Root Finding Techniques (Site not responding. Last check: 2007-10-08)
		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)

	CSI 610 CHAPTER 4 PROBLEM 7 (Site not responding. Last check: 2007-10-08)
		I wrote one matlab routine to perform the secant method against this function and a second matlab routine to plot the function.
		Setting initial estimates close to this 'zero' should yield convergance yet the secant method performs erratically against this equation.
		As can be observed, the secant method has a poor time finding the root.
	www.science.gmu.edu /~prowane/csi801/ch4p7.html (224 words)

	The plot of the value of f(x) vs
		The secant, bisection and falsi method seem to be equivalent as far is the number of iterations are concerned.
		[1,4]) in about three instances in the secant method, but assuming that the function is given by the same expression even outside the range, the value of
		We see that apart from the secant method, all the other methods converge rather well.
	www.rit.edu /~ssm8867/data/numericalmethods/hw2.htm (396 words)

	Homework Six (Site not responding. Last check: 2007-10-08)
		When it says to ``confirm your analysis by implementing each of the schemes...'' are we supposed to use one of the methods we developed in the last problem (i.e.
		A: Since you expect all the different methods to have different convergence rates, you have to call FZERO one time for each method.
		Q:Please explain why Newton's method (and finding roots) is appropriate for this problem and not the fixed-point iteration method (and finding convergence at x=2) on page 228.
	www.rpi.edu /~lvovy/Fall2005/node41.html (922 words)

	Secant Method (Site not responding. Last check: 2007-10-08)
		This module demonstrates the secant method for solving a nonlinear equation in one dimension.
		The successive steps of the secant method are then carried out sequentially by repeatedly clicking on NEXT or on the currently highlighted step.
		At each iteration of the secant method, the approximating secant line at the current points is drawn, the next approximate solution is taken to be the intersection of the secant line with the x axis, and the process is then repeated.
	www.cse.uiuc.edu /eot/modules/nonlinear_eqns/Secant (223 words)

	C# Tutorial: Numerical Methods
		The secant method is one of the simplest methods for solving algebraic equations.
		The simplest method to solve ODE is to do a Taylor expansion, which is called Euler's method.
		Runge-Kutta methods with a variable step size are often used in practice since they converge faster than fixed size methods.
	csharpcomputing.com /Tutorials/Lesson16.htm (419 words)

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