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Evaluation schemes in the ring of quaternionic polynomials

BIT Numerical Mathematics volume 58, pages 51–72 (2018)Cite this article

Abstract

In this paper we focus on computational aspects associated with polynomial problems in the ring of one-sided quaternionic polynomials. The complexity and error bounds of quaternion arithmetic are considered and several evaluation schemes are analyzed from their complexity point of view. The numerical stability of generalized Horner’s and Goertzel’s algorithms to evaluate polynomials with quaternion floating-point coefficients is addressed. Numerical tests illustrate the behavior of the algorithms from the point of view of performance and accuracy.

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Notes

  1. In the context of this paper, a flop is any of the four elementary arithmetic operations \(+\), −, \(\times \), \(\div \).

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Acknowledgements

The authors would like to thank an anonymous referee for the valuable and constructive suggestions.

Author information

Authors and Affiliations

  1. CMAT - Centre of Mathematics and DMA - Department of Mathematics and Applications, University of Minho, Braga, Portugal

    M. Irene Falcão & Fernando Miranda

  2. DMA - Department of Mathematics and Applications, University of Minho, Braga, Portugal

    Ricardo Severino

  3. NIPE - Economics Politics Research Unit and DMA - Department of Mathematics and Applications, University of Minho, Braga, Portugal

    M. Joana Soares

Authors
  1. M. Irene Falcão
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  2. Fernando Miranda
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  3. Ricardo Severino
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  4. M. Joana Soares
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Corresponding author

Correspondence to M. Irene Falcão.

Additional information

Communicated by Lars Eldén.

Research at CMAT was financed by Portuguese Funds through FCT - Fundação para a Ciência e a Tecnologia, within the Project UID/MAT/00013/2013. Research at NIPE was carried out within the funding with COMPETE reference number POCI-01-0145-FEDER-006683 (UID/ECO/03182/2013), with the FCT/MEC’s (Fundação para a Ciência e a Tecnologia, I.P.) financial support through national funding and by the ERDF through the Operational Programme on “Competitiveness and Internationalization—COMPETE 2020” under the PT2020 Partnership Agreement.

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Falcão, M.I., Miranda, F., Severino, R. et al. Evaluation schemes in the ring of quaternionic polynomials. Bit Numer Math 58, 51–72 (2018). https://doi.org/10.1007/s10543-017-0667-8

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  • DOI: https://doi.org/10.1007/s10543-017-0667-8

Keywords

  • Quaternions
  • Polynomial evaluation
  • Error analysis

Mathematics Subject Classification

  • 65Y20
  • 11R52
  • 12Y05

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