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 ?
One of the best root-finding methods for solving nonlinear scalar equation f(x) = 0 is Newton's iteration method.
Nonlinear system (103) can be solved by Newton's iteration.
However, for A = 2, 4, we have to start Newton's iteration with the following data:
are a coupled-mode solution of (18) obtained by using Newton's iteration with K = 0.
Caption: Figure 1: Vector soliton of system (1) obtained after 10 Newton's iterations and [delta] = 0, [[sigma].
To do this we first note that Newton's iteration for the given function is
For instance, in order to solve the equation t = [alpha]exp(t), Newton's iteration is [t.
The aim of this section is to show that Newton's iteration is appropriate.
Newton's iteration for AREs can be reformulated as a one-step iteration rewriting it such that the next iteration is computed directly from the Lyapunov equation in Step 2 of Algorithm 1.
The following Newton's iteration was employed to furnish a numerical approximation to the inverse,
A good initial guess (necessary for the Newton's iteration to converge) can be obtained by solving the initial value problem
MN] was then used as an initial guess for the Newton's iteration (2.