Halley's method

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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 iterative and its rate of convergence is cubic.

1 Method
2 Derivation
3 Cubic convergence
4 See also
5 External links
6 References

[edit] Method

Like any root-finding method, 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:

$x_{n+1} = x_n - \frac {2 f(x_n) f'(x_n)} {2 {[f'(x_n)]}^2 - f(x_n) f''(x_n)}$

beginning with an initial guess x₀.

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

$| x_{n+1} - a | \le K \cdot {| x_n - a |}^3$ , 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.

[edit] Derivation

Consider the function

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

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

$x_{n+1} = x_n - \frac {g(x_n)} {g'(x_n)}$

with

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

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

[edit] 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:

$0 = f(a) = f(x_n) + f'(x_n) (a - x_n) + \frac{f''(x_n)}{2} (a - x_n)^2 + \frac{f'''(\xi)}{6} (a - x_n)^3$

and also

$0 = f(a) = f(x_n) + f'(x_n) (a - x_n) + \frac{f''(\eta)}{2} (a - x_n)^2,$

where ξ and η are numbers lying between a and x_n. Multiply the first equation by $2 f'(x_n)\!$ and subtract from it the second equation times $f''(x_n) (a - x_n)\!$ to give:

$\begin{align} 0 = 2 f(x_n) f'(x_n) & + 2 [f'(x_n)]^2 (a - x_n) + f'(x_n) f''(x_n) (a - x_n)^2 + \frac{f'(x_n) f'''(\xi)} {3} (a - x_n)^3 \\ & - f(x_n) f''(x_n) (a - x_n) - f'(x_n) f''(x_n) (a - x_n)^2 - \frac{f''(x_n) f''(\eta)} {2} (a - x_n)^3. \end{align}$

Canceling $f'(x_n) f''(x_n) (a - x_n)^2\!$ and re-organizing terms yields:

$\begin{align} 0 = 2 f(x_n) f'(x_n) &+ \big(2 [f'(x_n)]^2 - f(x_n) f''(x_n) \big) (a - x_n) \\ &+ \left( \frac{f'(x_n) f'''(\xi)} {3} - \frac{f''(x_n) f''(\eta)} {2} \right) (a - x_n)^3. \end{align}$

Put the second term on the left side and divide through by $2[f'(x n)] 2 - f (x n) f''(x n)$ to get:

$a - x_n = \frac{- 2 f(x_n) f'(x_n)}{2 [f'(x_n)]^2 - f(x_n) f''(x_n)} - \frac{2 f'(x_n) f'''(\xi) - 3 f''(x_n) f''(\eta)} {6(2 [f'(x_n)]^2 - f(x_n) f''(x_n))} (a - x_n)^3.$

Thus:

$a - x_{n+1} = - \frac{2 f'(x_n) f'''(\xi) - 3 f''(x_n) f''(\eta)} {12 [f'(x_n)]^2 - 6 f(x_n) f''(x_n)} (a - x_n)^3.$

The limit of the coefficient on the right side as x_n approaches a is:

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

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.

[edit] See also

Newton's method
Taylor's theorem
Householder's method

[edit] External links

Eric W. Weisstein, Halley's method at 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)

[edit] References

T.R. Scavo and J.B. Thoo, On the geometry of Halley's method. American Mathematical Monthly, volume 102 (1995), number 5, pages 417–426.
This article began as a translation from the equivalent article in French Wikipedia, retrieved 22 January 2007.

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Category: Root-finding algorithms