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Polynomiography via Ishikawa and Mann Iterations

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Advances in Visual Computing (ISVC 2012)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 7431))

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Abstract

The aim of this paper is to present some modifications of the complex polynomial roots finding visualization process. In this paper Ishikawa and Mann iterations are used instead of the standard Picard iteration. The name polynomiography was introduced by Kalantari for that visualization process and the obtained images are called polynomiographs. Polynomiographs are interesting both from educational and artistic points of view. By the use of different iterations we obtain quite new polynomiographs that look aestheatically pleasing comparing to the ones from standard Picard iteration. As examples we present some polynomiographs for complex polynomial equation z 3 − 1 = 0, permutation and doubly stochastic matrices. We believe that the results of this paper can inspire those who may be interested in created automatically aesthetic patterns. They also can be used to increase functionality of the existing polynomiography software.

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References

  1. Berinde, V.: Iterative Approximation of Fixed Points, 2nd edn. Springer, Heidelberg (2007)

    MATH  Google Scholar 

  2. Ishikawa, S.: Fixed Points by a New Iteration Method. Proceedings of the American Mathematical Society 44(1), 147–150 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  3. Kalantari, B.: Polynomiography: From the Fundamental Theorem of Algebra to Art. Leonardo 38(3), 233–238 (2005)

    Article  Google Scholar 

  4. Kalantari, B.: Polynomial Root-Finding and Polynomiography. World Scientific, Singapore (2009)

    Google Scholar 

  5. Kalantari, B.: Alternating Sign Matrices and Polynomiography. The Electronic Journal of Combinatorics 18(2), 1–22 (2011)

    MathSciNet  Google Scholar 

  6. Mandelbrot, B.: The Fractal Geometry of Nature. W.H. Freeman and Company, New York (1983)

    Google Scholar 

  7. Mann, W.R.: Mean value methods in iteration. Proceedings of the American Mathematical Society 4, 506–510 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  8. Minc, H.: Nonnegative Matrices. John Wiley & Sons, New York (1988)

    MATH  Google Scholar 

  9. Prasad, B., Katiyar, K.: Fractals via Ishikawa Iteration. In: Balasubramaniam, P. (ed.) ICLICC 2011. CCIS, vol. 140, pp. 197–203. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  10. Singh, S.L., Jain, S., Mishra, S.N.: A New Approach to Superfractals. Chaos, Solitons and Fractals 42(5), 3110–3120 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Susanto, H., Karjanto, N.: Newton’s Method’s Basins of Attraction Revisited. Applied Mathematics and Computation 215(3), 1084–1090 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Computer Science, University of Silesia, Poland

    Wiesław Kotarski, Krzysztof Gdawiec & Agnieszka Lisowska

Authors
  1. Wiesław Kotarski
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  2. Krzysztof Gdawiec
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  3. Agnieszka Lisowska
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Nevada, 89557, Reno, NV, USA

    George Bebis

  2. NASA Ames Research Center, 94035, Moffett Field, CA, USA

    Richard Boyle

  3. Lawrence Berkeley National Laboratory, 94720, Berkeley, CA, USA

    Bahram Parvin

  4. Desert Research Institute, 89512, Reno, NV, USA

    Darko Koracin

  5. Department of Computer Science, University of California at Irvine, 92697-3435, Irvine, CA

    Charless Fowlkes

  6. Eastman Kodak Company, 14650-2102, Rochester, NY, USA

    Sen Wang

  7. Department of Computer Science and Engineering, University of Colorado at Denver, 80217, Denver, CO, USA

    Min-Hyung Choi

  8. VRVis Zentrum für Virtual Reality and Visualisierung, 1220, Vienna, Austria

    Stephan Mantler

  9. California Institute for Telecommunications and Information Technology, University of California, 92093, San Diego, La Jolla, CA, USA

    Jürgen Schulze

  10. KAUST Visualizatioin Core Lab., 23955-6900, Thurwal, Saudi Arabia

    Daniel Acevedo

  11. Stony Brook University, 11794-4400, NY, USA

    Klaus Mueller

  12. Argonne National Laboratory, 60439, IL, USA

    Michael Papka

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© 2012 Springer-Verlag Berlin Heidelberg

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Kotarski, W., Gdawiec, K., Lisowska, A. (2012). Polynomiography via Ishikawa and Mann Iterations. In: Bebis, G., et al. Advances in Visual Computing. ISVC 2012. Lecture Notes in Computer Science, vol 7431. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33179-4_30

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  • DOI: https://doi.org/10.1007/978-3-642-33179-4_30

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33178-7

  • Online ISBN: 978-3-642-33179-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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