Muller's Method

Generalizes the Secant Method of root finding by using quadratic 3-point interpolation

$\begin{displaymath} q\equiv {x_n-x_{n-1}\over x_{n-1}-x_{n-2}}. \end{displaymath}$

(1)

Then define

$\displaystyle A$	$\textstyle \equiv$	$\displaystyle qP(x_n)-q(1+q)P(x_{n-1})+q^2P(x_{n-2})$	(2)
$\displaystyle B$	$\textstyle \equiv$	$\displaystyle (2q+1)P(x_n)-(1+q)^2P(x_{n-1})+q^2P(x_{n-2})$
			(3)
$\displaystyle C$	$\textstyle \equiv$	$\displaystyle (1+q)P(x_n),$	(4)

and the next iteration is

$\begin{displaymath} x_{n+1} = x_n-(x_n-x_{n-1}){2C\over\max(B\pm\sqrt{B^2-4AC}\,)}. \end{displaymath}$

(5)

This method can also be used to find Complex zeros of Analytic Functions.

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, p. 364, 1992.