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Accurate Goertzel Algorithm: Error Analysis, Validations and Applications

Author

Listed:
  • Chuanying Li

    (College of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, China)

  • Peibing Du

    (Northwest Institute of Nuclear Technology, Xi’an 710024, China)

  • Kuan Li

    (School of Cyberspace Security, Dongguan University of Technology, Dongguan 523106, China)

  • Yu Liu

    (Northwest Institute of Nuclear Technology, Xi’an 710024, China)

  • Hao Jiang

    (College of Computer, National University of Defense Technology, Changsha 410073, China)

  • Zhe Quan

    (College of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, China)

Abstract

The Horner and Goertzel algorithms are frequently used in polynomial evaluation. Each of them can be less expensive than the other in special cases. In this paper, we present a new compensated algorithm to improve the accuracy of the Goertzel algorithm by using error-free transformations. We derive the forward round-off error bound for our algorithm, which implies that our algorithm yields a full precision accuracy for polynomials that are not too ill-conditioned. A dynamic error estimate in our algorithm is also presented by running round-off error analysis. Moreover, we show the cases in which our algorithms are less expensive than the compensated Horner algorithm for evaluating polynomials. Numerical experiments indicate that our algorithms run faster than the compensated Horner algorithm in those cases while producing the same accurate results, and our algorithm is absolutely stable when the condition number is smaller than 10 16 . An application is given to illustrate that our algorithm is more accurate than MATLAB ’s fft function. The results show that the relative error of our algorithm is from 10 15 to 10 17 , and that of the fft was from 10 12 to 10 15 .

Suggested Citation

  • Chuanying Li & Peibing Du & Kuan Li & Yu Liu & Hao Jiang & Zhe Quan, 2022. "Accurate Goertzel Algorithm: Error Analysis, Validations and Applications," Mathematics, MDPI, vol. 10(11), pages 1-19, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:11:p:1788-:d:822463
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    References listed on IDEAS

    as
    1. Graillat, Stef, 2018. "An accurate algorithm for evaluating rational functions," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 494-503.
    2. Jiang, Hao & Graillat, Stef & Barrio, Roberto & Yang, Canqun, 2016. "Accurate, validated and fast evaluation of elementary symmetric functions and its application," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 1160-1178.
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