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Topic: Newtons method

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Sir Isaac Newton

	Newton's method - Wikipedia, the free encyclopedia
		The idea of the method is as follows: one starts with a value which is reasonably close to the true zero, then replaces the function by its tangent (which can be computed using the tools of calculus) and computes the zero of this tangent (which is easily done with elementary algebra).
		Newton's method was described by Isaac Newton in De analysi per aequationes numero terminorum infinitas (written in 1669, published in 1711 by William Jones) and in De metodis fluxionum et serierum infinitarum (written in 1671, translated and published as Method of Fluxions in 1736 by John Colson).
		Newton's method was first published in 1685 in A Treatise of Algebra both Historical and Practical by John Wallis.
	en.wikipedia.org /wiki/Newtons_method (1640 words)

	Newton, Isaac (1642-1727) -- from Eric Weisstein's World of Scientific Biography
		Newton was forced to leave Cambridge when it was closed because of the plague, and it was during this period that he made some of his most significant discoveries.
		Newton was extremely sensitive to criticism, and even ceased publishing until the death of his arch-rival Hooke.
		Newton mathematized all of the physical sciences, reducing their study to a rigorous, universal, and rational procedure which marked the ushering in of the Age of Reason.
	www.treasure-troves.com /bios/Newton.html (1527 words)

	Newtons Method failed to converge in 100 iteration
		Re: Newtons Method failed to converge in 100 itera - Ogbeni, Tue, 7 Mar 2006, 1:56 p.m.
		Re: Newtons Method failed to converge in 100 itera - Joe, Fri, 10 Mar 2006, 12:28 a.m.
		Re: Newtons Method failed to converge in 100 itera - hagupta, Fri, 10 Mar 2006, 7:36 a.m.
	www.cfd-online.com /Forum/cfx_archive.cgi?read=14080 (211 words)

	Overview — 6. Functions — The DRM
		Methods may be invoked directly (used as functions), or indirectly through the invocation of a generic function.
		Methods added to a generic function must have parameter lists that are congruent with the generic function's parameter list.
		Methods created directly can be stored in module variables, passed as arguments to generic functions, stored in data structures, or immediately invoked.
	www.opendylan.org /books/drm/Functions_Overview (1135 words)

	GNU Scientific Library -- Reference Manual - Root Finding Algorithms using Derivatives
		Newton's method converges quadratically for single roots, and linearly for multiple roots.
		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.
	www.math.utah.edu /software/gsl/gsl-ref_407.html (438 words)

	Lecture 14
		The former application leads to a numerical procedure called "Newton's Method" which, for example, is a popular technique used by calculators to find square roots.
		Using Newton's Method, find the equation of a line tangent to the curve y = sin(x) that passes through the origin.
		Use Newton's Method to approximate to 6 decimal places the solution to the equation tan(x/4) = 1.
	www.math.dartmouth.edu /~m3cod/lecture14index.htm (318 words)

	Newton's Method
		Essentially, Newton's method requires the error surface to be close to quadratic, and its effectiveness is directly tied to the accuracy of this assumption.
		For this reason, Newton's method tends to give very rapid (``quadratic'') convergence in the last stages of iteration.
		Newton's method is derived as follows: The Taylor expansion of
	ccrma-www.stanford.edu /~jos/gradient/Newton_s_Method.html (191 words)

	Procedures as Returned Values
		Newton's method is the use of the fixed-point method we saw above to approximate a solution of the equation by finding a fixed point of the function f.
		In order to implement Newton's method as a procedure, we must first express the idea of derivative.
		Since Newton's method was itself expressed as a fixed-point process, we actually saw two ways to compute square roots as fixed points.
	mitpress.mit.edu /sicp/full-text/sicp/book/node25.html (1656 words)

	Re: Newtons Method failed to converge in 100 itera
		Re: Newtons Method failed to converge in 100 itera
		Re: Newtons Method failed to converge in 100 itera (Stone)
		Newtons Method failed to converge in 100 iteration - hagupta, Mon, 6 Mar 2006, 9:13 a.m.
	www.cfd-online.com /Forum/cfx_archive.cgi?read=14261 (225 words)

	[No title]
		Lin and Bairstow’s methods have proved to be the more popular techniques in control engineering since these methods extract complex conjugate root pairs quite easily which is of great importance in that field.
		The Secant and Bisection methods are generally ineffective when it comes to finding complex conjugate root pairs, while the Newton-Raphson method and Muller’s method use complex arithmetic which is awkward when using hand calculation but possible in a computer program.
		The method proves to be very good for extracting complex conjugate root pairs from real polynomials, and its quadratic convergence rate is also favourable compared to other methods.
	meltingpot.fortunecity.com /oxford/647/ra_compa.html (1302 words)

	Project One (Site not responding. Last check: 2007-10-23)
		The purpose of this project is to help you understand the principles behind Newton's method, and to use them to develop an "optimized" version.
		The project is to devise an implementation of Newton's method that combines the best features of both, so that is goes quickly to the nearest solution from the largest possible set of starting points.
		What you might do here is to write a loop that would go through the points on a fine grid, and run Newton's method for each of the two ways until you see what happens, and then record the result, The arrays of points can then be plotted giving you pictures of the sets.
	www.math.gatech.edu /~carlen/2507/lab/proj1.html (867 words)

	Homework Six (Site not responding. Last check: 2007-10-23)
		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.
		As I understand it, the fixed-point iteration that we worked with in the development of the convergence at x=2 is a seperate issue from finding the roots by means of Newton's method or a similar 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)

	newtons
		Moreover the quasi-Newton methods of the last two sections are able to effectively function as a second order method as they approach the solution – and they do not need the estimation of second derivatives.
		As mentioned before, for real design problems, where decisions are required to be made on the discrete nature of some of the variables, the existence of the first derivative, leave alone second derivatives is questionable.
		The Newton's method is the extension of the Newton-Raphson method for many variables.
	www.rit.edu /~pnveme/EMEM820n/Mod6_Numerical/mod6_content/mod6_sec2_newtons.html (159 words)

	Dr. Dobb's \| The Dylan Programming Language \| July 22, 2001
		Methods are defined with typed formal parameters and can be applied to arguments with either the same class as the defined parameters or to subclasses of the defined parameters.
		Defining a method automatically creates a new generic function with the same name, and the new method is attached to the new generic function.
		A new method is defined by creating a new, unique identifier, called a "key," for this method, having Scheme bind the new key to the method's equivalent lambda expression in the Scheme namespace, adding the parameter list and key to the appropriate generic function, and creating a new generic function if required.
	www.ddj.com /article/printableArticle.jhtml?articleID=184409404&dept_url= (2611 words)

	Dylan reference manual -- Function Defining Forms
		The method parameters are bound as lexical variables over the scope of the implicit-body.
		method is a method that accepts the arguments described by parameter-list and then executes the implicit-body.
		Bare methods may be called directly or they may be added to generic functions.
	core.federated.com /~jim/dirm/interim-24.html (853 words)

	MATH 5334 Home Page (Site not responding. Last check: 2007-10-23)
		It is well-known that the use of numerical methods for the analysis, simulation, and design of engineering processes and systems has been increasing at a rapid rate.
		Therefore, this course is intended to better prepare future engineers, as well as to assist practicing engineers, in understanding the fundamentals of numerical methods, especially their application, limitations, and potentials.
		The objective will be to understand why the methods work, what type of errors to expect, and when an application might lead to difficulties.
	www.math.ttu.edu /~padhu/math5334e_s05.html (682 words)

	Secant Method
		This is the secant method; we still use a chord on the curve to estimate the next approximation to the root, but now the chord is drawn between the last two points on the curve, rather than the most recent bracketing pair.
		Although we have derived this method in a fairly ad hoc way, it turns out that the secant method is one of the best methods to use for ``difficult' roots providing we can abandon the guarantee of the bisection method.
		We discuss this further after describing one more method, which is essentially the secant method taken to the limit in the sense of the calculus.
	www.maths.abdn.ac.uk /~igc/tch/mx3015/notes/node109.html (302 words)

	[No title]
		To us e this worksheet first choose a function, then the number of itteratio ns you want to be done, then give a starting point for Newtons Method.
		See how Newton's me thod approximates the zero by playing with the animation buttons.
		I h ope this gives some understanding of Newton's method, and of some of t he interesting and fun things Maple is capable of.
	www.math.tamu.edu /~freeman/DBFNewtonsMethod.mws (195 words)

	Newton's Method
		Newton's method can be generalized to a system of nonlinear equations by expanding
		To assure convergence, it requires an initial guess which is close enough to the desired solution, and a step size which is small enough.
		IMSL subroutines based on Newton's method, like NEQNF and NEQNJ, can be used to solve in practice nonlinear systems of equations.
	www.math.vt.edu /people/renardym/class_home/nova/bifs/node50.html (82 words)

	SUB Göttingen - Dissertationen - Arghanoun, Ghazaleh: Random Iterations of Subhyperbolic Relaxed Newton'ss Methods
		The family of the natural iterations of a subhyperbolic relaxed Newton’s method of a complex polynomial devides the Riemann sphere into two disjoint sets: the nonempty, perfect, nowhere dense, connected and locally connected Julia set with Lebesgue measure zero, and the open Fatou set with stable simply connected components.
		We study first the dynamics of a family of random iterations of the relaxed Newton’s methods of the polynomials in a small neighborhood of a given fixed polynomial whose relaxed Newton’s method is subhyperbolic.
		Finally, we give an application of the methods discussed in the previous chapters to the case when the relaxed Newton’s method of a given polynomial has at least one parabolic periodic point in its Julia set.
	webdoc.sub.gwdg.de /diss/2005/arghanoun (580 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		Considering A and B as functions of p and q we have a nonlinear system A(p.q)=0 B(p,q)=0 which is solved by Newtons method (without damping and control of descent).
		The algorithm you mentioned is Laguerres method, which is globally convergent, can deliver complex zeroes from real initial guesses.
		This is a really good method, but once in the complex plane you need complex arithmetic.
	www.math.niu.edu /~rusin/known-math/01_incoming/bairstow (508 words)

	New root-seeking algorithm
		Thus, the Bisection method is a special case of the Quartile method, where the new guess is systematically taken as the middle of the root-bracketing interval.
		Often, complex root-seeking implementation use a combination of Newton (or similar/better methods) with he Bisection as a backup plan in case Newtons method yield guesses that stray away.
		The method of False-Position is the next higherup root-bracketing method that uses the function values as weights.
	www.hpmuseum.org /cgi-sys/cgiwrap/hpmuseum/archv015.cgi?read=83105 (889 words)

	Math 1507 Lab Projects (Site not responding. Last check: 2007-10-23)
		One of the Java applets to be used with this project: This one determines which root a particular starting point converges to using Newtons method.
		One of the Java applets to be used with this project: This one produces the sequence of succesive approximations obtained from a given starting point using Newton's method.
		The Maple worksheet to be used with this project: This Maple worksheet was written by Evelyn Sander and is a thorough introduction to graphing with Maple.
	www.math.gatech.edu /~carlen/1507/lab (124 words)

	X2 Numerical Maths Home Page
		A check the method is consistent and find the order of the theoretical method (you need pen and paper for this bit!).
		Can you find a method of testing the convergence of a series in a text book (or searching on the web) that can be applied to this summation?
		This was done in the programme for the trapezoidal method.
	www.staff.city.ac.uk /~ra359/X2NumMaths (1291 words)

	University of Pittsburgh: Department of Mathematics (Site not responding. Last check: 2007-10-23)
		Newton's Method - known to all from Calculus I - is an iterative technique for finding an approximation to a single zero of a function.
		However, since the value of the root obtained by Newton's Method is approximate all of the coefficients of q also are approximate - sometimes the roots of the approximate q are wildly different from the remaining roots of p.
		The purpose of our project was to read a recent paper of Hubbard et al concerning the geometry and topology of Newton's Method when applied to polynomials.
	www.math.pitt.edu /spring2002-zeros.html (551 words)

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