Newton's method

Also found in: Wikipedia.

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 ?
The Newton observer applies Newton's method to find the solution.
The local order of convergence of Newton's method is two and it is optimal with two function evaluations per iterative step.
Besides, the outage capacity of DAS in the presence of imperfect CSI is also derived, and a Newton's method based practical iterative algorithm is proposed to find the accurate outage capacity.
We recall that for any nonconstant meromorphic function u, the Newton's method of finding the zeros of u consists of iterating the function f defined by
On radial distributions of Julia sets of Newton's method of solutions of complex differential equations .
To find the maximum of L(w, [mu]), Newton's method is used to optimize L(w, [mu]).
The study of (Krishnamurthy and Murthy, 1983) describes a fast iterative scheme based on the Newton's method for finding the reciprocal of finite segment p-adic numbers.
Keywords-: fixed point method, Jungck iterative method, Nonlinear functional equation, Newton's method.
The topics include optimality conditions for unconstrained optimization, the gradient method, Newton's method, convex functions, optimality conditions for linearly constrained problems, and duality.
The most popular method for solving a differentiable system of nonlinear equations G(x) = 0 is Newton's method [8], which require calculating, at each iteration, the Jacobian matrix of G.
Most notable are implicit time-stepping schemes and Newton's method for solving nonlinear equation systems.