Halley's method

Edmond Halley was an English mathematician who introduced the method now called by his name. 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:

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

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 x_n satisfy:

${\displaystyle |x_{n+1}-a|\leq K\cdot {|x_{n}-a|}^{3},{\text{ for some }}K>0.}$

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

The following alternative formulation shows the similarity between Halley's method and Newton's method. The expression ${\displaystyle f(x_{n})/f'(x_{n})}$ is computed only once, and it is particularly useful when ${\displaystyle f''(x_{n})/f'(x_{n})}$ can be simplified:

${\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})-{\frac {f(x_{n})}{f'(x_{n})}}{\frac {f''(x_{n})}{2}}}}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}\left[1-{\frac {f(x_{n})}{f'(x_{n})}}\cdot {\frac {f''(x_{n})}{2f'(x_{n})}}\right]^{-1}.}$

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

Consider the function

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

Any root of f which is not a root of its derivative is a root of g; and any root 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

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

with

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

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

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:

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

and also

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

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

${\displaystyle {\begin{aligned}0&=2f(x_{n})f'(x_{n})+2[f'(x_{n})]^{2}(a-x_{n})+f'(x_{n})f''(x_{n})(a-x_{n})^{2}+{\frac {f'(x_{n})f'''(\xi )}{3}}(a-x_{n})^{3}\\&\qquad -f(x_{n})f''(x_{n})(a-x_{n})-f'(x_{n})f''(x_{n})(a-x_{n})^{2}-{\frac {f''(x_{n})f''(\eta )}{2}}(a-x_{n})^{3}.\end{aligned}}}$

Canceling ${\displaystyle f'(x_{n})f''(x_{n})(a-x_{n})^{2}}$ and re-organizing terms yields:

${\displaystyle 0=2f(x_{n})f'(x_{n})+\left(2[f'(x_{n})]^{2}-f(x_{n})f''(x_{n})\right)(a-x_{n})+\left({\frac {f'(x_{n})f'''(\xi )}{3}}-{\frac {f''(x_{n})f''(\eta )}{2}}\right)(a-x_{n})^{3}.}$

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

${\displaystyle 2[f'(x_{n})]^{2}-f(x_{n})f''(x_{n})}$

to get:

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

Thus:

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

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

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

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:

${\displaystyle |a-x_{n+1}|\leq K|a-x_{n}|^{3}}$

which is what was to be proved.

To summarize,

${\displaystyle \Delta x_{i+1}={\frac {3(f'')^{2}-2f'f'''}{12(f')^{2}}}(\Delta x_{i})^{3}+O[\Delta x_{i}]^{4},\qquad \Delta x_{i}\triangleq x_{i}-a.}$[4]

• 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)