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 ?
The polynomials are easy to code and fast to compute as only seven multiplications are required for each polynomial if factorized appropriately The Newton-Raphson method for inverting the projection converges quickly, with only a few iterations required.
T] This system of equations may then be solved using the Newton-Raphson method or a variant, given an initial iterate for the port-plane system variables,{^.
T] As before, this system of equations may then be solved using the Newton-Raphson method, given an initial iterate for the loop system variables, [{m}.
2) * ASAM = Accelerating convergence of Successive approximation method, NM = Newton-Raphson Method, SAM = Successive approximation method,
2003, Improving Newton-Raphson method for nonlinear equations by modified Adomian decomposition method, Applied Mathematics and Computation, 145:887-893.
6 is solved via the Newton-Raphson method since it exhibits terminal quadrature convergence and is conducive to the design sensitivity analysis to follow.
10 is substituted into any of the viscosity equations given in Table 1, we obtain a nonlinear equation in [eta], which must be solved numerically via an iterative procedure such as the Newton-Raphson method.
utilities and other energy organizations, including TEP, have analyzed their electrical power system by using a mathematical technique for calculating power flows, known as the Newton-Raphson method.
ext] The (N + 1) equations, obtained by applying Eq 20 to the (N + 1) nodes, are solved by using Newton-Raphson method [for additional details, see Fylchiron et al.
Then the Newton-Raphson method is used to solve the system of equations obtained by applying Eq 29 at each node.