# Newton's method

(redirected from Newton-Rapson Method)

## 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
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References in periodicals archive ?
1992; Rajcic, 1988) programs have been created in C++ programming language for Gauss-Seidel method, Fast-Decoupled method and Newton-Rapson method of Solution of Nonlinear Algebraic Equations.
2004) says that Newton-Rapson method has poor convergence for distribution radial networks and it is suggested to use alternative methods like Forward Sweeping method (Das et al.
In each iteration of Newton-Rapson method it is necessary to solve linear algebraic equations such as [A][x]=[b] where [A] is sparse, symmetrical and positive definite matrix.

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