What Root-Finding Algorithm Converges Faster Than Newton-Raphson?

[DuncanM]

Some time ago I saw a thread in which was mentioned a root-finding algorithm that converges twice as fast as the Newton-Raphson method. Newton-Raphson converges to a zero at a quadratic rate, and a poster pointed out that another algorithm converges to a zero at a quartic rate.

I have tried to find that thread, but cannot.

Anybody here know what algorithm I am talking about? It is for computing the roots of a function and converges to a solution at twice the rate of Newton-Raphson?

 

[Office_Shredder]

Googling 'root finding quartic convergence' gives this
http://idosi.org/wasj/wasj7(am)/14.pdf

It's just higher order Taylor series iterations

 

[DuncanM]

I think I found it, and here is an old thread in these forums about it:

https://www.physicsforums.com/showthread.php?t=409453

It is called Halley's Method.

I was mistaken; it actually offers cubic convergence, not quartic.
In any case, is still faster convergence than the Newton-Raphson Method.

Here are a couple other links with information about it:

http://en.wikipedia.org/wiki/Halley's_method

http://mathworld.wolfram.com/HalleysMethod.html

 

[Office_Shredder]

The paper I linked to does have a fourth order convergence