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 for a given polynomial . 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. However, for computer computation, more efficient methods are known, with which it is guaranteed to find all roots (see ) or all real roots (see Real-root isolation). This method is named in honour of Edmond Laguerre, a French mathematician. Definition The algorithm of the Laguerre method to find one root of a polynomial of degree is: Choose an initial guess For If is very small, exit the loop Calculate Calculate Calculate , where the sign is chosen to give the denominator with the larger absolute value, to avoid catastrophic cancellation. Set Repeat until a is small enough or if the maximum number of iterations has been reached. 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. Note however that deflation can lead to approximate factors that differ significantly from the corresponding exact factors. This error is least if the roots are found in the order of increasing magnitude. Derivation The fundamental theorem of algebra states that every nth degree polynomial can be written in the form such that where are the roots of the polynomial. If we take the natural logarithm of both sides, we find that Denote the derivative by and the negated second derivative by We then make what Acton calls a 'drastic set of assumptions', that the root we are looking for, say, is a certain distance away from our guess , and all the other roots are clustered together some distance away. If we denote these distances by and then our equation for may be written as and the expression for becomes Solving these equations for , we find that , where the square root of a complex number is chosen to produce larger absolute value of the denominator, or equivalently, to satisfy: , where denotes real part of a complex