Computation of the zeros of Laguerre–Sobolev polynomials

A new algorithm for computing all the zeros of Laguerre–Sobolev orthogonal polynomials, based on the Ehrlich–Aberth method, is described in this work. The Ehrlich–Aberth method is a Newton–like method, requiring, at each iteration, the evaluation of the polynomial and its derivative in the computed approximations of the zeros. The Laguerre–Sobolev polynomials are related to the classical Laguerre orthogonal polynomials by a four term recurrence relation, that allows to evaluate the former polynomials and their derivatives in a point. This relation can be then exploited in the Ehrlich–Aberth method. Laguerre–Sobolev polynomials exhibit a behavior similar to that of Laguerre polynomials: their values grow rapidly as their degrees increase, and overflow occurs in floating point arithmetic if their degree exceeds 170. In order to avoid overflow, novel recurrence relations are proposed to simultaneously compute the ratio between the Laguerre–Sobolev polynomials and the corresponding derivatives in a point. The proposed algorithm turns out to be very efficient and accurate, with O(n ² ) computational complexity and O(n) memory, where n is the degree of the polynomial. AMS Classification: 33C45 , 33C47 , 65H04