Newton's method

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Newton's method

[′nüt·ənz ‚meth·əd]
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

References in periodicals archive ?
To solve this nonlinear system one can introduce a supplementary variable and then apply a Newton method.
The basic calculation methods of periodic processes of nonlinear systems in time domain are such methods as pseudo viscosity, extrapolation [1], gradient [2] and quasi Newton method, or method which is focused on the definition of the vector of the initial values, corresponding to the established mode of the system.
For the most 6-DOF parallel robot, the nonlinear direct kinematic equations can be solved by the algorisms using Newton method [4, 11], homotopy method [5], neural network [6] and algebraic elimination [7].
The barrier method includes the Newton method for unconstrained minimization and a logarithmic barrier function.
Just look at the insignificant differences when comparing the results with those of Newton method.
In this work, we propose a nonsmooth quasi Newton method for solving the NCP using the system [[PHI].
WALKER, Choosing the forcing terms in an inexact Newton method, SIAM J.
Variants of the Gauss-Newton method, such as the projected Newton method [4], can be used (in Matlab[C]), and enable avoiding problems with derivatives of the cost function; at the same time it accelerates the convergence.
Also, by combining power means Newton method with the well-known Newton's method in a three step iteration process, we can get few existing sixth order methods as special cases of the present method and such method is p = 1 and -1 in [7, 3].
Newton Method is efficient if the guess is close to the root.
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).
Among them the one-step smoothing Newton method were proposed by [1] for NCP(F) and box constrained variational inequalities.