Laguerre's method

In numerical analysis, Laguerre's method is a root-finding algorithm tailored to polynomials.

In other words, Laguerre's method can be used to numerically solve the equation p(x) = 0 for a given polynomial p(x).

One of the most useful properties of this method is that it is, from extensive empirical study, very close to being a "sure-fire" method, meaning that it is almost guaranteed to always converge to some root of the polynomial, no matter what initial guess is chosen.

This method is named in honour of the French mathematician, Edmond Laguerre.

The algorithm of the Laguerre method to find one root of a polynomial p(x) of degree n is: If a root has been found, the corresponding linear factor can be removed from p. This deflation step reduces the degree of the polynomial by one, so that eventually, approximations for all roots of p can be found.

This error is least if the roots are found in the order of increasing magnitude.

The fundamental theorem of algebra states that every nth degree polynomial

If we take the natural logarithm of both sides, we find that Denote the logarithmic derivative by and the negated second derivative by We then make what Acton (1970)[page needed] calls a "drastic set of assumptions", that the root we are looking for, say,

we find that where in this case, the square root of the (possibly) complex number is chosen to produce largest absolute value of the denominator and make

denotes real part of a complex number, and

or where the square root of a complex number is chosen to have a non-negative real part.

this formula differs from the offset of the third order Halley's method by an error of

Even if the "drastic set of assumptions" does not work well for some particular polynomial p(x), then p(x) can be transformed into a related polynomial r for which the assumptions are viable; e.g. by first shifting the origin towards a suitable complex number w, giving a second polynomial q(x) = p(x − w), that give distinct roots clearly distinct magnitudes, if necessary (which it will be if some roots are complex conjugates).

The approximate root from Graeffe's method, can then be used to start the new iteration for Laguerre's method on r. An approximate root for p(x) may then be obtained straightforwardly from that for r. If we make the even more extreme assumption that the terms in

then Laguerre's method converges cubically whenever the initial guess,

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.

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,

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.