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Algorithms for accurate, validated and fast polynomial evaluation

Abstract

We survey a class of algorithms to evaluate polynomials with floating point coefficients and for computation performed with IEEE-754 floating point arithmetic. The principle is to apply, once or recursively, an error-free transformation of the polynomial evaluation with the Horner algorithm and to accurately sum the final decomposition. These compensated algorithms are as accurate as the Horner algorithm perforned inK times the working precision, forK an arbitrary positive integer. We prove this accuracy property with an a priori error analysis. We also provide validated dynamic bounds and apply these results to compute a faithfully rounded evaluation. These compensated algorithms are fast. We illustrate their practical efficiency with numerical experiments on significant environments. Comparing to existing alternatives theseK-times compensated algorithms are competitive forK up to 4, i.e., up to 212 mantissa bits.

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References

  1. [1]

    High-Precision Software Directory. http://crd.lbl.gov/~dhbailey/mpdist.

  2. [2]

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

  3. [3]

    J.W. Demmel, Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997.

  4. [4]

    S. Graillat, P. Langlois and N. Louvet, Compensated Horner scheme. Algebraic and Numerical Algorithms and Computer-Assisted Proofs, B. Buchberger, S. Oishi, M. Plum and S.M. Rump (eds.), Dagstuhl Seminar Proceedings, No. 05391, Internationales Begenungs-und Forschungszentrum (IBFI), Schloss Dagstuhl, Germany, 2006.

  5. [5]

    S. Graillat, P. Langlois and N. Louvet, Improving the compensated Horner scheme with a fused multiply and add. Proceedings of the 21st Annual ACM Symposium on Applied Computing, Vol. 2, Association for Computing Machinery, 2006, 1323–1327.

  6. [6]

    Y. Hida, X.S. Li and D.H. Bailey, Quad-double arithmetic: Algorithms, implementation, and application. 15th IEEE Symposium on Computer Arithmetic, N. Burgess and L. Ciminiera (eds.), IEEE Computer Society, 2001, 155–162.

  7. [7]

    N.J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd edition. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2002.

  8. [8]

    C.M. Hoffmann, G. Park, J.-R. Simard and N.F. Stewart, Residual iteration, and accurate polynomial evaluation for shape-interrogation applications. Proceedings of the 9th ACM Symposium on Solid Modeling and Applications, 2004, 9–14.

  9. [9]

    IEEE Standards Committee 754, IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Standard 754–1985. Institute of Electrical and Electronics Engineers, Los Alamitos, CA, USA, 1985, Reprinted in SIGPLAN Notices,22 (1987), 9–25.

  10. [10]

    D.E. Knuth, The Art of Computer Programming: Seminumerical Algorithms, Vol. 2, 3rd edition. Addison-Wesley, Reading, MA, USA, 1998.

  11. [11]

    P. Langlois, More accuracy at fixed precision. J. Comp. Appl. Math.,162 (2004), 57–77.

  12. [12]

    P. Langlois and N. Louvet, How to ensure a faithful polynomial evaluation with the compensated Horner algorithm. 18th IEEE International Symposium on Computer Arithmetic, P. Kornerup and J.-M. Muller (eds.), IEEE Computer Society, 2007, 141–149.

  13. [13]

    P. Langlois and N. Louvet, More instruction level parallelism explains the actual efficiency of compensated algorithms. Technical Report hal-00165020, DALI Research Project, HALCCSD 2007.

  14. [14]

    P. Langlois and N. Louvet, Compensated Horner algorithm inK times the working precision. RNC-8, Real Numbers and Computer Conference, J. Brugera and M. Daumas (eds.), Santiago de Compostela, Spain, 2008.

  15. [15]

    X.S. Li, J.W. Demmel, D.H. Bailey, G. Henry, Y. Hida, J. Iskandar, W. Kahan, S.Y. Kang, A. Kapur, M.C. Martin, B.J. Thompson, T. Tung and D.J. Yoo, Design, implementation and testing of extended and mixed precision BLAS. ACM Trans. Math. Software,28 (2002), 152–205.

  16. [16]

    The MPFR Library (version 2.2.1). Available at http://www.mpfr.org.

  17. [17]

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

  18. [18]

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

  19. [19]

    S.M. Rump, T. Ogita and S. Oishi, Accurate floating-point summation, Part I: Faithful rounding. SIAM J. Sci. Comput.,31 (2008), 189–224.

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Author information

Correspondence to Stef Graillat.

Additional information

This work has been prepared while the authors were members of DALI at ELIAUS, laboratory of Université de Perpignan, and partly funded by the ANR Project EVA-Flo (ANR-BLAN06-2-135670 2006).

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Cite this article

Graillat, S., Langlois, P. & Louvet, N. Algorithms for accurate, validated and fast polynomial evaluation. Japan J. Indust. Appl. Math. 26, 191–214 (2009). https://doi.org/10.1007/BF03186531

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Key words

  • polynomial evaluation
  • compensated algorithm
  • floating-point arithmetic
  • IEEE-754