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

Abstract

Abstract-ZusammenfassungThe Range of Values of a Circular Complex Polynomial over a Circular Complex Interval. Two Methods to find a small circular interval that encloses the range of values of a circular interval polynomial over a circular interval are developed. The first method uses Bernstein polynomials over the sides of a regular polygon enclosing the domain, the other method uses a mean value property of curves in the complex plane. They are then compared to each other as well as to the result obtained from the Hornerscheme and the true range. Der Wertebereich eines Intervallpolynoms mit Kreisen als Koeffizienten.Es werden zwei Methoden entwickelt, einen m6glichst kleinen Kreisbereich zu finden, der den Wertebereich eines Kreisintervallpolynoms auf einem gegebenen Kreisbereich einschliegt. Die erste Methode beniitzt Bernsteinpolynome auf den Seiten eines umschriebenen reguliiren Polygons, die andere eine Mittelwerteigeuschaft yon Kurven in der komplexen Ebene. Die beiden werden untereinander und auch mit dem Hornerschema und dem wirklichen Wertebereich verglichen.

Bounds for the range of a complex polynomial over a rectangular region

“…As we have mentioned in Subsection 3.1 and after Theorem 4, these enclosures can be tightened by degree elevation of the Bernstein expansion as well as by subdivision (preferred). The test polynomials p i , i = 1, 4, 5, are taken from [20], p i , i = 2, 3, from [10], and p i , i = 6, 7, from [21]. The timings presented in Table 6 clearly document the superiority of Method III.…”

“…The first paper in this regard was [19] in which the range of a complex polynomial over the interval [0, 1] is enclosed. Rokne [20] and Grassmann and Rokne [10] extended this result to range enclosures over rectangles and discs in the complex plane. Furthermore, they considered enclosures for the range of complex polynomials with coefficients which are not exactly known but can located within rectangles or discs.…”

“…The enclosures themselves are listed in Table 9 in Appendix A. In [20], the range is considered over z 1 and in [10] over the disc of radius 1 centered at the origin. In both references, also an approximation of the contour of the respective range is presented which allows to judge upon the quality of the enclosure.…”