Halley's method

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. C² 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 the 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. There do exist multidimensional versions of this method.

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:

:$x\_\{n+1\}\; =\; x\_n\; -\; frac\; \{2\; f(x\_n)\; f\text{'}(x\_n)\}\; \{2\; \{\; [f\text{'}(x\_n)]\; \}^2\; -\; f(x\_n)\; f"(x\_n)\}$

beginning with an initial guess "x"₀.

If ƒ is a thrice continuously differentiable function and "a" is a zero of ƒ but not of its derivative, then, in a neighborhood of "a", the iterates "x"_"n" satisfy:

:$|\; x\_\{n+1\}\; -\; a\; |\; le\; K\; cdot\; ^3,\; ext\{\; for\; some\; \}K\; >\; 0.!$

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

Derivation

Consider the function

:$g(x)\; =\; frac\; \{f(x)\}\; \{sqrt\{|f\text{'}(x)|.$

Any root of ƒ which is "not" a root of its derivative is a root of "g"; and any root of "g" is a root of ƒ. Applying Newton's method to "g" gives

:$x\_\{n+1\}\; =\; x\_n\; -\; frac\; \{g(x\_n)\}\; \{g\text{'}(x\_n)\}$

with

:$g\text{'}(x)\; =\; frac\; \{2\; \{\; [f\text{'}(x)]\; \}^2\; -\; f(x)\; f"(x)\}\; \{2\; f\text{'}(x)\; sqrt\{|f\text{'}(x)|,$

and the result follows. Notice that if ƒ'("c") = 0, then one cannot apply this at "c" because "g"("c") would be undefined.

Cubic convergence

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 "x"_"n" is in that neighborhood. Then Taylor's theorem implies:

:$0\; =\; f(a)\; =\; f(x\_n)\; +\; f\text{'}(x\_n)\; (a\; -\; x\_n)\; +\; frac\{f"(x\_n)\}\{2\}\; (a\; -\; x\_n)^2\; +\; frac\{f"\text{'}(xi)\}\{6\}\; (a\; -\; x\_n)^3$

and also

:$0\; =\; f(a)\; =\; f(x\_n)\; +\; f\text{'}(x\_n)\; (a\; -\; x\_n)\; +\; frac\{f"(eta)\}\{2\}\; (a\; -\; x\_n)^2,$

where ξ and η are numbers lying between "a" and "x"_"n". Multiply the first equation by $2\; f\text{'}(x\_n)!$ and subtract from it the second equation times $f"(x\_n)\; (a\; -\; x\_n)!$ to give:

:

Canceling $f\text{'}(x\_n)\; f"(x\_n)\; (a\; -\; x\_n)^2!$ and re-organizing terms yields:

:

Put the second term on the left side and divide through by $2\; [f\text{'}(x\_n)]\; ^2\; -\; f(x\_n)\; f"(x\_n)$ to get:

:$a\; -\; x\_n\; =\; frac\{-\; 2\; f(x\_n)\; f\text{'}(x\_n)\}\{2\; [f\text{'}(x\_n)]\; ^2\; -\; f(x\_n)\; f"(x\_n)\}-\; frac\{2\; f\text{'}(x\_n)\; f"\text{'}(xi)\; -\; 3\; f"(x\_n)\; f"(eta)\}\; \{6(2\; [f\text{'}(x\_n)]\; ^2\; -\; f(x\_n)\; f"(x\_n))\}\; (a\; -\; x\_n)^3.$

Thus:

:$a\; -\; x\_\{n+1\}\; =\; -\; frac\{2\; f\text{'}(x\_n)\; f"\text{'}(xi)\; -\; 3\; f"(x\_n)\; f"(eta)\}\; \{12\; [f\text{'}(x\_n)]\; ^2\; -\; 6\; f(x\_n)\; f"(x\_n)\}\; (a\; -\; x\_n)^3.$

The limit of the coefficient on the right side as "x"_"n" approaches "a" is:

:$-\; frac\{2\; f\text{'}(a)\; f"\text{'}(a)\; -\; 3\; f"(a)\; f"(a)\}\; \{12\; [f\text{'}(a)]\; ^2\}.$

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:

:$|a\; -\; x\_\{n+1\}|\; leq\; K\; |a\; -\; x\_n|^3$

which is what was to be proved.

References

* T.R. Scavo and J.B. Thoo, On the geometry of Halley's method. "American Mathematical Monthly", 102:5 (1995), pp. 417–426.
*This article began as a translation from [http://fr.wikipedia.org/w/index.php?title=Itération_de_Halley&oldid=11673690 the equivalent article in French Wikipedia] , retrieved 22 January 2007.

External links

*
* [http://math.fullerton.edu/mathews/n2003/Halley'sMethodMod.html Halley's Method by John H. Mathews]
* " [http://numbers.computation.free.fr/Constants/Algorithms/newton.html 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)