Laguerre's Method - Properties

Properties

If x is a simple root of the polynomial p, then Laguerre's method converges cubically whenever the initial guess x₀ is close enough to the root x. On the other hand, if x is a multiple root then the convergence is only linear. This is obtained with the penalty of calculating values for the polynomial and its first and second derivatives at each stage of the iteration.

A major advantage of Laguerre's method is that it is almost guaranteed to converge to some root of the polynomial no matter where the initial approximation is chosen. This is in contrast to other methods such as the Newton-Raphson method which may fail to converge for poorly chosen initial guesses. It may even converge to a complex root of the polynomial, because of the square root being taken in the calculation of a above may be of a negative number. This may be considered an advantage or a liability depending on the application to which the method is being used. Empirical evidence has shown that convergence failure is extremely rare, making this a good candidate for a general purpose polynomial root finding algorithm. However, given the fairly limited theoretical understanding of the algorithm, many numerical analysts are hesitant to use it as such, and prefer better understood methods such as the Jenkins-Traub method, for which more solid theory has been developed. Nevertheless, the algorithm is fairly simple to use compared to these other "sure-fire" methods, easy enough to be used by hand or with the aid of a pocket calculator when an automatic computer is unavailable. The speed at which the method converges means that one is only very rarely required to compute more than a few iterations to get high accuracy.