The nearest complex polynomial with a zero in a given complex domain

Abstract

Given a univariate complex polynomial f and a closed complex domain D , whose boundary C is a curve parameterized by a piecewise rational function, we propose two computational algorithms for finding a univariate complex polynomial f ̃ such that f ̃ has a zero in D and the distance between f and f ̃ is minimal. Our approach is composed of two steps. First, in the case of D consisting of one point α , we give explicit formulas of f ̃ and the minimal distance in terms of α . Next, the case of a general closed domain D is considered by using the property that a nearest polynomial f ̃ has a zero on the boundary C . The curve C is parameterized piecewisely, and on each piece we search for the minimum of the distance between f and f ̃ . At this step we exploit the explicit formula of the minimal distance as a function of a point α . Then the global minimum and the nearest polynomial are obtained by comparing the piecewise minima. Some examples are presented: one of them confirms that the distance between a nearest complex polynomial and a given polynomial is less than that between a nearest real polynomial and the given polynomial.