Improvement on Aberth's method for choosing initial approximations to zeros of polynomials

Abstract

Concerns a method for choosing the initial approximations to the zeros of a polynomial. Methods for finding all the zeros of a polynomial simultaneously, which can be considered modified Newton Methods, have been reported by quite a few authors. The initial approximations for these simultaneous methods take much CPU-time to find all the zeros of a polynomial, since CPU time depends mostly on the initial distribution of zeros. A new method for choosing the initial approximations is proposed, which simplifies Aberth's method (Math. Comput., vol.27, p.339-44, 1973) for choosing the initial approximations, and reduces the CPU time to obtain initial approximations and find all the zeros of a polynomial. In this method, the quasi-standard deviation obtained by the degree and coefficients of a polynomial is defined, and this gives the initial approximations. A comparison between the method is made through numerical experiments. >