Halley's method

Summary

In numerical analysis, Halley's method is a root-finding algorithm used for functions of one real variable with a continuous second derivative. Edmond Halley was an English mathematician and astronomer who introduced the method now called by his name.

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

Halley's method exactly finds the roots of a linear-over-linear Padé approximation to the function, in contrast to Newton's method or the Secant method which approximate the function linearly, or Muller's method which approximates the function quadratically.[1]

Method

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Halley's method is a numerical algorithm for solving the nonlinear equation f(x) = 0. In this case, the function f has to be a function of one real variable. The method consists of a sequence of iterations:

 

beginning with an initial guess x0.[2]

If f is a three times continuously differentiable function and a is a zero of f but not of its derivative, then, in a neighborhood of a, the iterates xn satisfy:

 

This means that the iterates converge to the zero if the initial guess is sufficiently close, and that the convergence is cubic.[3]

The following alternative formulation shows the similarity between Halley's method and Newton's method. The expression   is computed only once, and it is particularly useful when   can be simplified:

 

When the second derivative is very close to zero, the Halley's method iteration is almost the same as the Newton's method iteration.

Derivation

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Consider the function

 

Any root r of f that is not a root of its derivative is a root of g (i.e.,   when  ), and any root r of g must be a root of f provided the derivative of f at r is not infinite. Applying Newton's method to g gives

 

with

 

and the result follows. Notice that if f ′(c) = 0, then one cannot apply this at c because g(c) would be undefined. [further explanation needed]

Cubic convergence

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Suppose a is a root of f but not of its derivative. And suppose that the third derivative of f exists and is continuous in a neighborhood of a and xn is in that neighborhood. Then Taylor's theorem implies:

 

and also

 

where ξ and η are numbers lying between a and xn. Multiply the first equation by   and subtract from it the second equation times   to give:

 

Canceling   and re-organizing terms yields:

 

Put the second term on the left side and divide through by

 

to get:

 

Thus:

 

The limit of the coefficient on the right side as xna is:

 

If we take K to be a little larger than the absolute value of this, we can take absolute values of both sides of the formula and replace the absolute value of coefficient by its upper bound near a to get:

 

which is what was to be proved.

To summarize,

 [4]

References

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  1. ^ Boyd, J. P. (2013). "Finding the Zeros of a Univariate Equation: Proxy Rootfinders, Chebyshev Interpolation, and the Companion Matrix". SIAM Review. 55 (2): 375–396. doi:10.1137/110838297.
  2. ^ Scavo, T. R.; Thoo, J. B. (1995). "On the geometry of Halley's method". American Mathematical Monthly. 102 (5): 417–426. doi:10.2307/2975033. JSTOR 2975033.
  3. ^ Alefeld, G. (1981). "On the convergence of Halley's method". American Mathematical Monthly. 88 (7): 530–536. doi:10.2307/2321760. JSTOR 2321760.
  4. ^ Proinov, Petko D.; Ivanov, Stoil I. (2015). "On the convergence of Halley's method for simultaneous computation of polynomial zeros". J. Numer. Math. 23 (4): 379–394. doi:10.1515/jnma-2015-0026. S2CID 10356202.
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  • Weisstein, Eric W. "Halley's method". MathWorld.
  • Newton's method and high order iterations, Pascal Sebah and Xavier Gourdon, 2001 (the site has a link to a Postscript version for better formula display)