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Computing

, Volume 33, Issue 1, pp 27–35 | Cite as

On the k-th root in circular arithmetic

  • Ljiljana Petković
  • M. Petković
Contributed Papers

Abstract

The representation of thek-th root of a complex circular intervalZ={c;r} is considered in this paper. Thek-th root is defined by the circular intervals which include the exact regionZ1/k={z:zk∈Z}. Two representations are given: (i) the centered inclusive disks\( \cup \{ c^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} ; \mathop {\max }\limits_{z \in Z} |z^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} - c^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} |\} \) and (ii) the diametrical inclusive disks with the diameter which is equal to the diameter of the regionZ1/k.

AMS Subject Classification

30E99 

Key words

Computational complex analysis circular interval arithmetic 

Über die k-te Wurzel in der Kreisarithmetik

Zusammenfassung

In diesem Artikel hat die Darstellung derk-ten Wurzel des komplexen KreisintervallsZ={c;r} übergelegt. Diek-te Wurzel wurde mit den Kreisintervallen definiert, welche das richtige GebietZ1/k={z:zk∈Z} einschließen. Zwei Darstellungen sind gegeben: (i) zentrierte inklusive Kreisscheiben\( \cup \{ c^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} ; \mathop {\max }\limits_{z \in Z} |z^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} - c^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} |\} \) und (ii) diametrische inklusive Kreisscheiben mit Diameter wie Diameter des GebietsZ1/k.

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References

  1. [1]
    Gargantini, I., Henrici, P.: Circular arithmetic and the determination of polynomial zeros. Numer. Math.18, 305–320 (1972).Google Scholar
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    Petković, M., Petković, Lj.: On a representation of thek-th root in complex circular arithmetic. In: Interval Mathematics 1980 (Nickel, K., ed.), pp. 473–479. New York: Academic Press 1980.Google Scholar
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    Petković, Lj., Petković, M.: The representation of complex circular functions using Taylor series. ZAMM61, 661–662 (1981).Google Scholar
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    Petković, Lj.: On two applications of Taylor series in circular complex arithmetic. Freiburger Intervall-Berichte2, 33–50 (1983).Google Scholar
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    Petković, M.: On a generalisation of the root iterations for polynomial complex zeros in circular interval arithmetic. Computing27, 37–55 (1981).Google Scholar
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    Rokne, J., Wu, T.: The circular complex centered form. Computing28, 17–30 (1982).Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Ljiljana Petković
    • 1
  • M. Petković
    • 1
  1. 1.Faculty of Mechanical EngineeringUniversity of NišNišYugoslavia

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