number, and is the complex conjugate of ; or , where the square root of a complex number is chosen to have a non-negative real part. For small values of this formula differs from the offset of the third order Halley's method by an error of , so convergence close to a root will be cubic as well. Note that, even if the 'drastic set of assumptions' does not work for some particular polynomial , can be transformed into a related polynomial for which the assumptions are correct, e.g. by shifting the origin towards a suitable complex number , giving , to give distinct roots distinct magnitudes if necessary (which it will be if some roots are complex conjugates), and then getting from by repeatedly applying the root squaring transformation used in Graeffe's method enough times to make the smaller roots significantly smaller than the largest root (and so, clustered in comparison); the approximate root from Graeffe's method can then be used to start the new iteration for Laguerre's method on . An approximate root for may then be obtained straightforwardly from that for . If we make the stronger assumption that the terms in corresponding to the roots are negligibly small in comparison to the term corresponding to the root , this leads to Newton's method. Properties If is a simple root of the polynomial , then Laguerre's method converges cubically whenever the initial guess is close enough to the root . On the other hand, if 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 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