Asymptotic Error Constant Definition
Contents |
Community Forums > Mathematics > Calculus > Dismiss Notice Join Physics Forums Today! The friendliest, high quality science and math community on the planet! Everyone who loves science is here! Rate of
Asymptotic Error Constant Newton’s Method
convergence and asymptotic error constant Jan 8, 2009 #1 azay In the context asymptotic standard error definition of root finding algorithms such as secant, regula falsi, bisection, Newton's method: In [tex] \lim_{n \to \infty} \frac{|x*-x_{n+1}|}{|x*-x_{n}|^{p}} = C [/tex]
What Is Asymptotic Error
I understand the meaning of the order p is the speed of convergence. For example, in Newton's method the order p = 2 and thus the number of correct significant digits is approximately newton method error formula doubled in each iteration step. But is there an intuitive meaning to be given to the asymptotic error constant C? What does this number mean? What is the difference between two methods that have the same order p, but for a different C? azay, Jan 8, 2009 Phys.org - latest science and technology news stories on Phys.org •Game over? Computer beats human champ in ancient Chinese game newton's method error bound •Simplifying solar cells with a new mix of materials •Imaged 'jets' reveal cerium's post-shock inner strength Feb 4, 2009 #2 Mathman85 As I understand it, if they are of the same order, the method with a smaller C will converge faster. Mathman85, Feb 4, 2009 (Want to reply to this thread? Log in or Sign up here!) Show Ignored Content Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook Have something to add? Struggles with the Continuum – Part 7 Orbital Precession in the Schwarzschild and Kerr Metrics Spectral Standard Model and String Compactifications Tetrad Fields and Spacetime Groups and Geometry Some Misconceptions on Indefinite Integrals LHC Part 4: Searching for New Particles and Decays Grandpa Chet’s Entropy Recipe Precession in Special and General Relativity Digital Camera Buyer’s Guide: Real Cameras Struggles with the Continuum – Conclusion Similar Discussions: Rate of convergence and asymptotic error constant Rate of Convergence (Replies: 3) Rate of convergence (Replies: 7) Constant rate of acceleration formula (Replies: 2) Rate of convergence for functions (Replies: 1) Choosing constants to make integral convergent (Replies: 5) Loading... Log in with Facebook Log in with Twitter Your name or email address: Do you
by Astozzia, Nadir Soualem All the versions of this article: [English] [francais]
Newton's method or Newton-Raphson method is an iterative numerical method used to solve f(x)=0 type equations. It relies onNewton's Method Error Analysis
the fixed-point method and on a particular function, g, related to the derivative error analysis for iterative methods of f. Definition Newton's method is a fixed-point method using the application : It can be easily inferred that looking
Newton's Method Error Estimate
for a fixed point for comes down to looking for a solution to the following equation Remember that, in order to look for the fixed point, we resort to an iterative algorithm https://www.physicsforums.com/threads/rate-of-convergence-and-asymptotic-error-constant.283588/ defined by the sequence The numerical scheme for Newton's method is
Geometrical interpretation The tangent to the curve f at the point has the following equation is nothing less than the abscissa of the point of intersection of this tangent with the -axis. Indeed We then set : Convergence of Newton's method Theorem. Let and such that and Then, there exists such that Newton's http://www.math-linux.com/mathematics/numerical-solution-of-nonlinear-equations/article/newton-s-method method converges for all Proof. By assumption, is continuous and . Therefore, there exists such that: The derivative of is defined by: By assumption, and . Consequently: Moreover, is continuous in since Explicitly writing the continuity of at in the interval yields to: that is Therefore, there exists such that Up to now, we have proved one of the assumptions for the fixed-point theorem. We now need to prove that the interval is -invariant. That is: By means of the mean value theorem, we show that there exists an element such that hence To sum up, we have proven that: et there exists a constant in ] 0,1[ such that . Via the fixed-point theorem, we can conclude that: the sequence defined by converges toward , the fixed point of . Order of convergence of Newton's method Theorem. If is continuous on an open set containing then, there exists such that the sequence defined by converges toward , the fixed point of , . Newton's method has a -order convergence. The convergence is quadratic. Proof. Similar to what we have seen previously, we show that there exisPlease note that Internet Explorer version 8.x will not be supported as of http://www.sciencedirect.com/science/article/pii/S0377042705005686 January 1, 2016. Please refer to this blog post for more information. Close ScienceDirectSign inSign in using your ScienceDirect credentialsUsernamePasswordRemember meForgotten username or password?Sign in via your institutionOpenAthens loginOther institution loginHelpJournalsBooksRegisterJournalsBooksRegisterSign inHelpcloseSign in using your ScienceDirect credentialsUsernamePasswordRemember meForgotten username or password?Sign in via your institutionOpenAthens loginOther institution login Download PDF method error Opens in a new window. Article suggestions will be shown in a dialog on return to ScienceDirect. Help Direct export Export file RIS(for EndNote, Reference Manager, ProCite) BibTeX Text RefWorks Direct Export Content Citation Only Citation and Abstract Advanced search JavaScript is disabled newton's method error on your browser. Please enable JavaScript to use all the features on this page. JavaScript is disabled on your browser. Please enable JavaScript to use all the features on this page. This page uses JavaScript to progressively load the article content as a user scrolls. Click the View full text link to bypass dynamically loaded article content. View full text Journal of Computational and Applied MathematicsVolume 196, Issue 2, 15 November 2006, Pages 347–357 Asymptotic error constants of cubically convergent zero finding methodsNaoki Osada Tokyo Woman's Christian University, Zempukuji, Suginamiku, Tokyo 167-8585, JapanReceived 15 January 2005, Revised 24 July 2005, Available online 2 December 2005AbstractThe Laguerre family of iteration functions for finding multiple zeros is considered. This family is algebraically equivalent to the multiple zero counterpart of Hansen–Patrick family. The asymptotic error constant for the Laguerre family is given. The magnitude of asymptotic error con