A root-finding algorithm which converges to a complex root 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)
			(11)

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

(12)

while all other roots are at the same distance , so

(13)

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

			(14)
			(15)

Solving these simultaneously for gives

(16)

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 (, , ).

Setting gives Halley's irrational formula.

Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., 1990.

Adams, D. A. "A Stopping Criterion for Polynomial Root Finding." Comm. ACM 10, 655-658, 1967.

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, pp. 365-366, 1992.

Ralston, A. and Rabinowitz, P. §8.9-8.13 in A First Course in Numerical Analysis, 2nd ed. New York: McGraw-Hill, 1978.

CITE THIS AS:

Weisstein, Eric W. "Laguerre's Method." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LaguerresMethod.html