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Accurate Horner methods in real and complex floating-point arithmetic

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Abstract

In this article, we derive accurate Horner methods in real and complex floating-point arithmetic. In particular, we show that these methods are as accurate as if computed in k-fold precision and then rounded into the working precision. When k is two, our methods are comparable or faster than the existing compensated Horner routines. When compared to multi-precision software, such as MPFR and MPC, our methods are significantly faster, up to k equal to eight, that is, up to 489 bits in the significand.

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References

  1. ANSI/IEEE: IEEE Standard for Binary Floating Point Arithmetic, std 754-2008 edn. IEEE, New York, NY (2008)

  2. ANSI/IEEE: IEEE Standard for Binary Floating Point Arithmetic, std 754-2019 edn. IEEE, New York, NY (2019)

  3. Cameron, T.R., Graillat, S.: On a compensated Ehrlich–Aberth method for the accurate computation of all polynomial roots. Electron. Trans. Numer. Anal. 55, 401–423 (2022)

    Article  MathSciNet  Google Scholar 

  4. Dekker, T.J.: A floating-point technique for extending the available precision. Numer. Math. 18, 224–242 (1971)

    Article  MathSciNet  Google Scholar 

  5. Enge, A., Gastineau, M., Théveny, P., Zimmermann, P.: MPC: A library for multiprecision complex arithmetic with exact rounding. Inria, 1.2.1 edn. (2021). http://mpc.multiprecision.org

  6. Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., Zimmermann, P.: MPFR: a multiple-precision binary floating-point library with correct rounding. ACM Trans. Math. Softw. 33(2), 13-es (2007)

    Article  MathSciNet  Google Scholar 

  7. Gill, S.: A process for the step-by-step integration of differential equations in an automatic digital computing machine. Proc. Camb. Philos. Soc. 47, 96–108 (1951)

    Article  MathSciNet  Google Scholar 

  8. Goldberg, D.: What every computer scientist should know about floating-point arithmetic. ACM Comput. Surv. 23, 5–48 (1991)

    Article  Google Scholar 

  9. Graillat, S., Louvet, N., Langlois, P.: Compensated Horner scheme. Technical report. Université de Perpignan Via Domitia (2005). https://www-pequan.lip6.fr/~jmc/polycopies/Compensation-horner.pdf

  10. Graillat, S., Ménissier-Morain, V.: Accurate summation, dot product and polynomial evaluation in complex floating point arithmetic. Inf. Comput. 216, 57–71 (2012)

    Article  MathSciNet  Google Scholar 

  11. Higham, N.J.: Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia (2002)

    Book  Google Scholar 

  12. Kahan, W.: Further remarks on reducing truncation errors. Commun. ACM 8, 40 (1965)

    Article  Google Scholar 

  13. Knuth, D.E.: The Art of Computer Programming: Seminumerical Algorithms, vol. 2. Addison-Wesley, Reading (1998)

    Google Scholar 

  14. Langlois, P., Louvet, N.: Compensated Horner algorithm in \(K\) times the working precision. Research Report (2008). https://hal.inria.fr/inria-00267077

  15. Møller, O.: Quasi double-precision in floating point addition. BIT 5, 37–50 (1965)

    Article  MathSciNet  Google Scholar 

  16. Nievergelt, Y.: Scalar fused multiply-add instructions produce floating-point matrix arithmetic provably accurate to the penultimate digit. ACM Trans. Math. Softw. 29(1), 27–48 (2003)

    Article  MathSciNet  Google Scholar 

  17. Ogita, T., Rump, S.M., Oishi, S.: Accurate sum and dot product. SIAM J. Sci. Comput. 26(6), 1955–1988 (2005)

    Article  MathSciNet  Google Scholar 

  18. Rump, S.M.: Inversion of extremely ill-conditioned. Jpn J. Ind. Appl. Math. 26, 249–277 (2009)

    Article  Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, Penn State Behrend, Erie, PA, USA

    Thomas R. Cameron

  2. CNRS, LIP6, Sorbonne Université, 75005, Paris, France

    Stef Graillat

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  1. Thomas R. Cameron
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  2. Stef Graillat
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Correspondence to Thomas R. Cameron.

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Communicated by Elisabeth Larsson

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This work was partly supported by the NuSCAP (ANR-20-CE48-0014) project of the French National Agency for Research (ANR).

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Cameron, T.R., Graillat, S. Accurate Horner methods in real and complex floating-point arithmetic. Bit Numer Math 64, 17 (2024). https://doi.org/10.1007/s10543-024-01017-w

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  • DOI: https://doi.org/10.1007/s10543-024-01017-w

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