Newton's method

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Newton's method

[′nüt·ənz ‚meth·əd]
(mathematics)
A technique to approximate the roots of an equation by the methods of the calculus.

Newton’s Method

 

a method of approximating a root x0 of the equation f(x) = 0; also called the method of tangents. In Newton’s method, the initial (“first”) approximation x = a1 is used to find a second, more accurate, approximation by drawing the tangent to the graph of y = f(x) at the point A[a1, f(a1)] up to the intersection of the tangent with the Ox-axis (see Figure 1). The point of intersection is x = a1f(a1)/f’(a1) and is adopted as the new value a2 of the root. By repeating this process as necessary, we can obtain increasingly accurate approximations a2, a3, … of the root x0 provided that the derivative f’(x) is monotonic and preserves its sign on the segment containing x0.

The error ε2 = x0a2 of the new value a2 is related to the old error ε1 = x0a1 by the formula

where f”(ξ) is the value of the second derivative of the function f(x) at some point ξ that lies between x0 and a 1. It is sometimes recommended that Newton’s method be used in conjunction with some other method, such as linear interpolation. Newton’s method allows generalizations, which makes it possible to use the method for solving equations f(x) = 0 in normed spaces, where F is an operator in such a space, in particular, for solving systems of equations and functional equations. This method was developed by I. Newton in 1669.

Newton's method

Newton-Raphson
References in periodicals archive ?
WALKER, Choosing the forcing terms in an inexact Newton method, SIAM J.
There are some traditional methods such as Newton method, Gauss method, Romberg method, and Simpson's method (Mi,1991), but they have many limitations like complexity in calculations, low precision, and their convergence is not guaranteed for higher order (Qu, 2010).
In areas with large resistivity and vertical structure contrast, the Gauss Newton method is significantly more accurate than the faster quasi-Newton method also available in RES2DINV (Dahlin and Loke 1998).
Kostic (2009) Determination of eigenvalues from middle spectrum of positive definite toeplitz matrix by newton method for characteristic polynomial Proceedings of 20th International DAAAM Symposium, ISBN 978-3-901509-70-4, ISSN 1726-9679, pp 192
We shall utilise a well known method, namely the classical Newton method for its simplicity and its second-order of convergence [5].
Algorithm 3(Modified Mamta Newton Method for Multiple Zeros(MMNM))
The Newton Method is essentially an educational system of cognitive social thinking communicated by an integrated educational method of coaching, modeling social acceptable behaviors and communication skills with daily self, peer and instructional reviews.
Newton Method with holder Derivatives, Numerical Functional Analysis and optimization, 25(5): 379-395.
As it is pointed out, the paper presents new modified Newton method which is based on the Taylor expansion of the characteristic polynomial and respectively of the even and odd characteristic polynomial.
However, what mainly interests us are the relations between the two sequences generated by the predictor-corrector (PC) homotopy continuation method and the damped Newton method, since in actual implementation these two sequences generally deviate somewhat from the continuous paths.
In primal-dual methods [12,13], steps are generated by applying a perturbed Newton method to the following system: