Halley's method

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Description:
In numerical analysis, Halley’s method is a root-finding algorithm used for functions of one real variable with a continuous second derivative, i.e. C<sup>2</sup> functions. It is named after its inventor Edmond Halley, who also discovered Halley's Comet.

The algorithm is second in the class of Householder's methods, right after Newton's method. Like the latter, it produces iteratively a sequence of approximations to the root; their rate of convergence to the root is cubic. Multidimensional versions of this method exist.

Method

Like any root-finding method, Halley’s method is a numerical algorithm for solving the nonlinear equation ƒ(x) = 0. In this case, the function ƒ has to be a function of one real variable. The method consists of a sequence of iterations:

<math>x_ = x_n - frac </math>


beginning with an initial guess x<sub>0</sub>.

If ƒ is a three times continuously differentiable function and a is a zero of &fnof; but not of its derivative, then, in a neighborhood of a, the iterates x<sub>n</sub> satisfy:

<math>| x_ - a | le K cdot ^3,textK > 0.!</math>


This means that the iterates converge to the zero if the initial guess is sufficiently close, and that the convergence is cubic.

The following alternative formulation shows the similarity between Halley’s method and Newton’s method. The expression <math>f(x_n)/f'(x_n)</math> is computed only once, and it is particularly...
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