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 ?
They cover topics in real and complex number complexity theory; the real solving of algebraic varieties with intrinsic complexity; the complexity and geometry of numerically solving polynomial systems; multiplicity hunting and approximating multiple roots of polynomial systems; the intrinsic complexity of elimination problems in effective algebraic geometry; and Newton iteration, conditioning, and zero counting.
1] since this is done once and for all, after which the fixed point of the modified Newton iteration procedure is sought by successive approximations.
In particular, we give a fast oracle based on Newton iteration.
It consists of an algorithm which performs the Newton iteration in O([n.
2 by using the one-level INB-NE algorithm (INB-NE1), in which the bad component near the shock is eliminated with an inner Newton iteration.
Inexact Newton iteration [MATHEMATICAL EXPRESSION NOT # iter.
Then, one step of the Newton iteration for a given starting matrix can be implemented as shown in Algorithm 1.
A popular approach is the Newton-Krylov procedure, where the outer Newton iteration uses an inner iterative procedure of Krylov type to solve the linear system (3.
2) with the classical Newton iteration which is defined by the linear (4 x 4) system
The vectors involved in this update do not come from the Newton iteration but they are generated using information from the CG iteration.
The only issue now is, how to define, for each Newton iteration, the subdomain problems on both fine and coarse meshes and the coarse mesh problem.
A standard choice for improving the convergence rate of the gradient projection iteration is to implement, instead, the projected Newton iteration [3, 10].