Editing Laguerre's method (section)

==Properties==
[[Image:Attraction zones of Laguerre's.png|250px|right|thumb|Attraction zones of Laguerre's method for the polynomial <math>p(x) = x^4 + 2 x^3 + 3 x^2 + 4 x + 1.</math>]]

If {{mvar|x}} is a simple root of the polynomial <math>p(x),</math> then Laguerre's method converges [[rate of convergence|cubically]] whenever the initial guess, <math>x^{(0)},</math> is close enough to the root <math>x_1.</math> On the other hand, when <math>x_1 </math> is a [[multiple root]] convergence is merely linear, 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's method|Newton–Raphson method]] and [[Stephensen's method]], which notoriously fail to converge for poorly chosen initial guesses. Laguerre's method may even converge to a complex root of the polynomial, because the radicand of the square root may be of a negative number, in the formula for the correction, <math>a,</math> given above – manageable so long as complex numbers can be conveniently accommodated for the calculation. This may be considered an advantage or a liability depending on the application to which the method is being used.

Empirical evidence shows 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 a default, and prefer better understood methods such as the [[Jenkins–Traub algorithm]], for which more solid theory has been developed and whose limits are known.

The algorithm is fairly simple to use, compared to other "sure-fire" methods, and simple enough for hand calculation, aided by a pocket calculator, if a computer is not available. 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.