the iteration is xn+1 = xn – f(xn)/f ′(xn), n = 0,1,2,…
where x0 is an approximation to the solution. For the system f(x) = 0, f = (f1,f2,…,fn)T, x = (x1,x2,…,xn)T,
the iteration takes the mathematical form xn+1 = xn – J(xn)–1f(xn), n = 0,1,2,…
where J(x) is the n×n matrix whose i,jth element is fi(x)/xj
In practice each iteration is carried out by solving a system of linear equations. Subject to appropriate conditions the iteration converges quadratically (ultimately an approximate squaring of the error occurs). The disadvantage of the method is that a constant recalculation of J may be too time-consuming and so the method is most often used in a modified form, e.g. with approximate derivatives. Since Newton's method is derived by a linearization of f(x), it is capable of generalization to other kinds of nonlinear problems, e.g. boundary-value problems (see ordinary differential equations).
"Newtons method." A Dictionary of Computing. . Encyclopedia.com. (December 11, 2018). https://www.encyclopedia.com/computing/dictionaries-thesauruses-pictures-and-press-releases/newtons-method
"Newtons method." A Dictionary of Computing. . Retrieved December 11, 2018 from Encyclopedia.com: https://www.encyclopedia.com/computing/dictionaries-thesauruses-pictures-and-press-releases/newtons-method
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:
Modern Language Association
The Chicago Manual of Style
American Psychological Association
- Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
- In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.