Approximating Complex Polynomial Zeros: Modified Weyl's Quadtree Construction and Improved Newton's Iteration

Abstract

We propose a new algorithm for the classical and still practically important problem of approximating zeros zj of an nth degree polynomial p(x) within error bound 2−bmaxj|zj|. The algorithm uses O((n2logn)log(bn)) arithmetic operations and comparisons for approximating all the n zeros and O((knlogn)log(bn)) for approximating the k zeros lying in a fixed domain (disc or square) and isolated from the other zeros. Unlike the previous fast algorithms of this kind, the new algorithm has its simple elementary description, is convenient for practical implementation, and allows the users to adapt the computational precision to the current level of approximation achieved in the process of computing and ultimately to the requirements to the output precision for each zero of p(x). The algorithm relies on our novel versions of Weyl's quadtree construction and Newton's iteration.