Traub’s accelerating generator of iterative root-finding methods

An accelerating generator of iterative methods for finding multiple roots, based on Traub’s differential–difference recurrence relation, is presented. It is proved that this generator yields an iteration function of order r+1r+1 starting from arbitrary iteration function of order rr. In this way, it is possible to construct various iterative formulas of higher order for finding single roots of nonlinear equations and all simple or multiple roots of algebraic polynomials, simultaneously. For demonstration, two iterative methods of the fourth order in ordinary (real or complex) arithmetic and an iterative method in interval arithmetic are presented.