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 ?
Using Taylor Series and Newton Iteration methods, functions beyond the scope of polynomials can also be computed by DNA circuits built upon our architecture.
In second place, we compute numerically some vector solitons of system (1) in forms (2) and (3) using a Newton iteration, combined with a collocation-spectral strategy to discretize the corresponding soliton equations.
Therefore, the Newton iteration algorithm is usually utilized to improve the process.
Newton iteration method converges fast and the complexity can be controlled just by the number of iterations [10].
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.
He, Improvement of Newton iteration method, International Journal of Nonlinear Sciences and Numerical Simulation, 1(2000), 239-240.
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.
This can be formally shown by noting that the Newton iteration for a cubic function applied to the conjugate [bar.
In particular, we give a fast oracle based on Newton iteration.
As the temperature effects increases, this problem becomes more expressed, and Newton iteration fails.
To find a zero of y = f(x), the well-known Newton iteration may be used.
Zhao, "Shearlet-wavelet regularized semismooth newton iteration for image restoration," Mathematical Problems in Engineering, vol.