The range of values of a complex polynomial over a complex interval

Abstract

--ZusammenfassungThe Range of Values of a Complex Polynomial Over a Complex Interval. We discuss algorithms for the computation of the range of values of a complex interval polynomial over a complex interval. The mathematical results needed are based upon a result by Rivlin [7] valid for the range of values of a complex polynomial over the line segment [0,1]. In the present work we extend his results to an arbitrary line segment in the complex plane. Based upon these results we then generate algorithms suitable for both line segments and rectangular regions in the complex plane executed in rectangular complex interval arithmetic. The algorithms are then tested on a variety of complex interval polynomials and compared both to the true rafige of values and to the values obtained by the Horner scheme. Der Wertevorrat eines komplexen Polynoms in einen komplexen Interval AnalysisInterval analysis is now a well established subject [31, [5] and here we will only introduce some notation. The set of closed intervals on the real line R will be denoted by I (R). The set of polynomials with coefficients in 1 (R) will be denoted by I P (R).The complex plane is denoted by C. If zl, z 2 ~ C, re (zl) <_ re (Zz) and im (zl) _< im (z2) then we call [zl, z21 a rectangular complex interval. The set of rectangular complex intervals is denoted by R C (C) (see [81, [-10]). If z s C and r s R then [z, r] is called a circular complex interval. The set of circular complex intervals is denoted by C I (C) (see [31, [4]).The elementary operations on elements from 1 (R) and 1P (R) are described in [51.The operations on elements from RI (C) are described in [101 and the operations 0010-485X/79/0022/0153/$ 03.40

The range of values of a circular complex polynomial over a circular complex interval

“…He also develops a method for a real polynomial over a real interval using the mean-value theorem. In [11] we extended his results using Bernstein polynomials to an arbitrary line segment in the complex plane. In this paper we extend the use of the mean-value theorem to arbitrary curves in the complex plane.…”

“…In [11] we extended these results to computation of the range of values of a rectangular complex interval polynomial over a rectangular complex interval. In this paper we discuss specific algorithms for the computation of the range of values of a circular complex interval polynomial over a circular complex interval.…”

“…As the number of sides increases, the quality of the approximation to the hull of the range will improve. We will quote a couple of relevant results from [11] as well as develop an algorithm. To find the best enclosing circle is a problem and we develop an algorithm for a good approximate solution.…”

“…Based on the above outline of the Bernstein method and on [11] we give a short description of the algorithms used to calculate the range of values of a complex polynomial using the Bernstein method.…”

Guest Editors' Preface

“…Rivlin [23] proved (linear) convergence of the bounds when the degree of the expansion is elevated and considered the case of complex polynomial coefficients. In a series of papers including [24,25,26], Rokne extended the results to (real and complex) interval polynomials. Lane and Riesenfeld [16] introduced subdivision, which exhibits quadratic convergence of the bounds, see, e.g., [10,12].…”