A root-finding algorithm which converges to a complexroot from any starting position. To motivate the formula, consider
an th
order polynomial and its derivatives,

(1)

(2)

(3)

(4)

Now consider the logarithm and logarithmic derivatives of

(5)

(6)

(7)

(8)

(9)

(10)

Now make "a rather drastic set of assumptions" that the root being sought is a distance from the current best guess, so

(11)

while all other roots are at the same distance , so

(12)

for ,
3, ...,
(Acton 1990; Press et al. 1992, p. 365). This allows and to be expressed in terms of and as

(13)

(14)

Solving these simultaneously for gives

(15)

where the sign is taken to give the largest magnitude for the denominator.

To apply the method, calculate for a trial value , then use as the next trial value, and iterate until becomes sufficiently small. For example, for the polynomial
with starting point , the algorithmic converges to
the real root very quickly as (, , ).