Laguerre's Method

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


Now consider the logarithm and logarithmic derivatives of P_n(x)


Now make "a rather drastic set of assumptions" that the root x_1 being sought is a distance a from the current best guess, so


while all other roots are at the same distance b, so


for i=2, 3, ..., n (Acton 1990; Press et al. 1992, p. 365). This allows G and H to be expressed in terms of a and b as


Solving these simultaneously for a gives


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

To apply the method, calculate a for a trial value x, then use x-a as the next trial value, and iterate until a becomes sufficiently small. For example, for the polynomial 4x^3+3x^2+2x+1 with starting point x_0=-1.0, the algorithmic converges to the real root very quickly as (-1.0, -0.58113883008419, -0.60582958618827).

Setting n=2 gives Halley's irrational formula.

See also

Halley's Irrational Formula, Halley's Method, Newton's Method, Root

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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.

Referenced on Wolfram|Alpha

Laguerre's Method

Cite this as:

Weisstein, Eric W. "Laguerre's Method." From MathWorld--A Wolfram Web Resource.

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