Generalizations of Laguerre’s Method: Higher Order Methods

Abstract

Laguerre’s method is an efficient and reliable method for finding zeros of polynomials and certain other functions. A new derivation and motivation of Laguerre’s method is given, which allows it to be included in a class of methods as general as methods of order three or more based on direct generalized Hermite or hyperosculatory interpolation. Members of this new class share with Laguerre’s method the property of being globally convergent to zeros of polynomials with only real zeros and have the same order of convergence at simple zeros as the classic methods based on generalized Hermite interpolation. Methods of order 4 and 3.303 are investigated and numerical results indicate that for large $(100 \times 100)$ eigenvalue problems the method of order 3.303 is as efficient and reliable as Laguerre’s method.