Skip to main content
SpringerLink
Log in

Pseudo-Newton Method with Fractional Order Derivatives

  • Conference paper
  • First Online:
Computational Science – ICCS 2022 (ICCS 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13351))

Included in the following conference series:

  • 974 Accesses

Abstract

Recently, the pseudo-Newton method was proposed to solve the problem of finding the points for which the maximal modulus of a given polynomial over the unit disk is attained. In this paper, we propose a modification of this method, which relies on the use of fractional order derivatives. The proposed modification is evaluated twofold: visually via polynomiographs coloured according to the number of iterations, and numerically by using the convergence area index, the average number of iterations and generation time of polynomiographs. The experimental results show that the fractional pseudo-Newton method for some fractional orders behaves better in comparison to the standard algorithm.

W. Kotarski—Independent Researcher.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Akgül, A., Cordero, A., Torregrosa, J.: A fractional Newton method with \(2\alpha \)th-order of convergence and its stability. Appl. Math. Lett. 98, 344–351 (2019). https://doi.org/10.1016/j.aml.2019.06.028

    Article  MathSciNet  MATH  Google Scholar 

  2. Cordero, A., Girona, I., Torregrosa, J.: A variant of Chebyshev’s method with \(3\alpha \)th-order of convergence by using fractional derivatives. Symmetry 11(8), Article 1017 (2019). https://doi.org/10.3390/sym11081017

  3. Erfanifar, R., Sayevand, K., Esmaeili, H.: On modified two-step iterative method in the fractional sense: some applications in real world phenomena. Int. J. Comput. Math. 97(10), 2109–2141 (2020). https://doi.org/10.1080/00207160.2019.1683547

    Article  MathSciNet  MATH  Google Scholar 

  4. Gdawiec, K., Kotarski, W.: Polynomiography for the polynomial infinity norm via Kalantari’s formula and nonstandard iterations. Appl. Math. Comput. 307, 17–30 (2017). https://doi.org/10.1016/j.amc.2017.02.038

    Article  MathSciNet  MATH  Google Scholar 

  5. Gdawiec, K., Kotarski, W., Lisowska, A.: Visual analysis of the Newton’s method with fractional order derivatives. Symmetry 11(9), Article ID 1143 (2019). https://doi.org/10.3390/sym11091143

  6. Gdawiec, K., Kotarski, W., Lisowska, A.: Newton’s method with fractional derivatives and various iteration processes via visual analysis. Numer. Algorithms 86(3), 953–1010 (2020). https://doi.org/10.1007/s11075-020-00919-4

    Article  MathSciNet  MATH  Google Scholar 

  7. Kalantari, B.: A geometric modulus principle for polynomials. Am. Math. Mon. 118(10), 931–935 (2011). https://doi.org/10.4169/amer.math.monthly.118.10.931

    Article  MathSciNet  MATH  Google Scholar 

  8. Kalantari, B.: A necessary and sufficient condition for local maxima of polynomial modulus over unit disc. arXiv:1605.00621 (2016)

  9. Kalantari, B., Andreev, F., Lau, C.: Characterization of local optima of polynomial modulus over a disc. Numer. Algorithms 3, 1–15 (2021). https://doi.org/10.1007/s11075-021-01208-4

    Article  MATH  Google Scholar 

  10. Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons Inc., New York (1993)

    MATH  Google Scholar 

  11. Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Amsterdam (1993)

    MATH  Google Scholar 

  12. Usurelu, G., Bejenaru, A., Postolache, M.: Newton-like methods and polynomiographic visualization of modified Thakur process. Int. J. Comput. Math. 98(5), 1049–1068 (2021). https://doi.org/10.1080/00207160.2020.1802017

    Article  MathSciNet  MATH  Google Scholar 

  13. Ypma, T.: Historical development of Newton-Raphson method. SIAM Rev. 37(4), 531–551 (1995). https://doi.org/10.1137/1037125

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computer Science, University of Silesia, Bȩdzińska 39, 41-200, Sosnowiec, Poland

    Krzysztof Gdawiec & Agnieszka Lisowska

  2. Katowice, Poland

    Wiesław Kotarski

Authors
  1. Krzysztof Gdawiec
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Agnieszka Lisowska
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wiesław Kotarski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Krzysztof Gdawiec .

Editor information

Editors and Affiliations

  1. Brunel University London, London, UK

    Derek Groen

  2. University of Amsterdam, Amsterdam, The Netherlands

    Clélia de Mulatier

  3. AGH University of Science and Technology, Krakow, Poland

    Maciej Paszynski

  4. University of Amsterdam, Amsterdam, The Netherlands

    Valeria V. Krzhizhanovskaya

  5. University of Tennessee at Knoxville, Knoxville, TN, USA

    Jack J. Dongarra

  6. University of Amsterdam, Amsterdam, The Netherlands

    Peter M. A. Sloot

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Gdawiec, K., Lisowska, A., Kotarski, W. (2022). Pseudo-Newton Method with Fractional Order Derivatives. In: Groen, D., de Mulatier, C., Paszynski, M., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M.A. (eds) Computational Science – ICCS 2022. ICCS 2022. Lecture Notes in Computer Science, vol 13351. Springer, Cham. https://doi.org/10.1007/978-3-031-08754-7_22

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-08754-7_22

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-08753-0

  • Online ISBN: 978-3-031-08754-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation