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

Abstract: Given a univariate complex polynomial and a closed complex domain , whose boundary is a curve parameterized by a piecewise rational function, we propose two computational algorithms for finding a univariate complex polynomial such that has a zero in and the distance between and is minimal. Our approach is composed of two steps. First, in the case of consisting of one point , we give explicit formulas of and the minimal distance in terms of . Next, the case of a general closed domain is considered by using the property that a nearest polynomial has a zero on the boundary . The curve is parameterized piecewisely, and on each piece we search for the minimum of the distance between and . At this step we exploit the explicit formula of the minimal distance as a function of a point . 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. [Copyright &y& Elsevier]