More accuracy at fixed precision. (English) Zbl 1037.65046
Summary: Several different techniques and software intend to improve the accuracy of results computed in a fixed finite precision. Here we focus on the CENA method that processes an automatic correction of the first-order effect of the rounding errors the computation generates. This method provides a corrected result and a bound of the residual error for a class of algorithms we identify. We present the main features of the CENA method and illustrate its interests and limitations with examples.
MSC:
| 65G50 | Roundoff error |
Keywords:
Automatic correction; CENA method; Accuracy; Finite precision; Floating; point arithmetic; rounding errorsReferences:
| [1] | D.H. Bailey, A Fortran-90 double-double library, 2001. Available at URL:; D.H. Bailey, A Fortran-90 double-double library, 2001. Available at URL: |
| [2] | Blackford, L. S.; Demmel, J.; Dongarra, J.; Duff, I.; Hammarling, S.; Henry, G.; Heroux, M.; Kaufman, L.; Lumsdaine, A.; Petitet, A.; Pozo, R.; Remington, K.; Whaley, R. C., An updated set of basic linear algebra subprograms (BLAS), ACM Trans. Math. Software, 28, 2, 135-151 (2002) · Zbl 1070.65520 |
| [3] | Bohlender, G., Floating-point computation of functions with maximum accuracy, IEEE Trans. Comput., C-26, 7, 621-632 (1977) · Zbl 0405.65021 |
| [4] | K. Briggs, Doubledouble floating point arithmetic, 1998. Available at URL:; K. Briggs, Doubledouble floating point arithmetic, 1998. Available at URL: |
| [5] | J.-M. Chesneaux, Stochastic arithmetic and CADNA software, Habilitation à diriger des recherches, Université P. et M. Curie, Paris, France, November 1995 (in French).; J.-M. Chesneaux, Stochastic arithmetic and CADNA software, Habilitation à diriger des recherches, Université P. et M. Curie, Paris, France, November 1995 (in French). |
| [6] | J.-M. Chesneaux, S. Guilain, J. Vignes, La bibliothèque CADNA: présentation et utilisation, Manual, Laboratoire d’Informatique de Paris 6, Université P. et M. Curie, Paris, France, November 1996. Available at URL:; J.-M. Chesneaux, S. Guilain, J. Vignes, La bibliothèque CADNA: présentation et utilisation, Manual, Laboratoire d’Informatique de Paris 6, Université P. et M. Curie, Paris, France, November 1996. Available at URL: |
| [7] | G. Corliss, C. Faure, A. Griewank, L. Hascoet, U. Naumann (Eds.), Automatic Differentiation Algorithms: From Simulation to Optimization, Computer and Information Science, Springer, New York, 2001.; G. Corliss, C. Faure, A. Griewank, L. Hascoet, U. Naumann (Eds.), Automatic Differentiation Algorithms: From Simulation to Optimization, Computer and Information Science, Springer, New York, 2001. |
| [8] | Dahlquist, G.; Björck, Å., Numerical Methods (1974), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ, USA, translated by Ned Anderson |
| [9] | Daumas, M.; Langlois, P., Additive symmetriesthe non-negative case, Theoret. Comput. Sci., 291, 2, 143-157 (2003) · Zbl 1064.68002 |
| [10] | Dekker, T. J., A floating-point technique for extending the available precision, Numer. Math., 18, 224-242 (1971) · Zbl 0226.65034 |
| [11] | J.W. Demmel, Recent progress in high accuracy and high performance linear algebra algorithms, SIAM Annual Meeting, May 1999. Available at URL:; J.W. Demmel, Recent progress in high accuracy and high performance linear algebra algorithms, SIAM Annual Meeting, May 1999. Available at URL: |
| [12] | J.W. Demmel, Y. Hida, Accurate floating point summation, Report UCB/CSD-02-1180, Department of Computer Science, University of California, May 2002.; J.W. Demmel, Y. Hida, Accurate floating point summation, Report UCB/CSD-02-1180, Department of Computer Science, University of California, May 2002. · Zbl 1061.65039 |
| [13] | Document for the Basic Linear Algebra Subprograms (BLAS) standard, February 2001. URL:; Document for the Basic Linear Algebra Subprograms (BLAS) standard, February 2001. URL: |
| [14] | Griewank, A., Evaluating Derivatives. Principles and Techniques of Algorithmic Differentiation (2000), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA, USA · Zbl 0958.65028 |
| [15] | Y. Hida, X.S. Li, D.H. Bailey, Quad-double arithmetic: algorithms, implementation, and application, Technical Report LBNL-46996, Department of Computer Science, University of California, Berkeley, CA, USA, October 2000.; Y. Hida, X.S. Li, D.H. Bailey, Quad-double arithmetic: algorithms, implementation, and application, Technical Report LBNL-46996, Department of Computer Science, University of California, Berkeley, CA, USA, October 2000. |
| [16] | Hida, Y.; Li, X. S.; Bailey, D. H., Quad-double arithmetic: algorithms, implementation, and application, (Burgess, N.; Ciminiera, L., 15th IEEE Symposium on Computer Arithmetic (2001), Institute of Electrical and Electronics Engineers: Institute of Electrical and Electronics Engineers New York), 155-162 |
| [17] | Higham, N. J., The accuracy of floating point summation, SIAM J. Sci. Comput., 14, 4, 783-799 (1993) · Zbl 0788.65053 |
| [18] | Higham, N. J., Accuracy and Stability of Numerical Algorithms (1996), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA, USA · Zbl 0847.65010 |
| [19] | IEEE Computer Society, New York, IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Standard 754-1985, reprinted in SIGPLAN Notices 22(2) 1987 (1985) 9-25.; IEEE Computer Society, New York, IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Standard 754-1985, reprinted in SIGPLAN Notices 22(2) 1987 (1985) 9-25. |
| [20] | Kahan, W., Further remarks on reducing truncation errors, Comm. ACM, 8, 1, 40 (1965) |
| [21] | Kulisch, U. W.; Miranker, W. L., Computer Arithmetic in Theory and in Practice (1981), Academic Press: Academic Press New York · Zbl 0437.65046 |
| [22] | Langlois, P., Automatic linear correction of rounding errors, BIT, 41, 3, 515-539 (2001) · Zbl 0988.65036 |
| [23] | Langlois, P.; Nativel, F., Reduction and bounding of the rounding error in floating point arithmetic, C.R. Acad. Sci. Paris, Sér. 1, 327, 781-786 (1998) · Zbl 0918.68037 |
| [24] | Langlois, P.; Nativel, F., When automatic linear correction of rounding errors is exact, C.R. Acad. Sci. Paris, Sér. 1, 328, 543-548 (1999), printer erratum in 328 (1999) 829 · Zbl 0936.65060 |
| [25] | Li, X. S.; Demmel, J. W.; Bailey, D. H.; Henry, G.; Hida, Y.; Iskandar, J.; Kahan, W.; Kang, S. Y.; Kapur, A.; Martin, M. C.; Thompson, B. J.; Tung, T.; Yoo, D. J., Design, implementation and testing of extended and mixed precision BLAS, ACM Trans. Math. Software, 28, 2, 152-205 (2002) · Zbl 1070.65523 |
| [26] | Linnainmaa, S., Error linearization as an effective tool for experimental analysis of the numerical stability of algorithms, BIT, 23, 346-359 (1983) · Zbl 0515.65040 |
| [27] | Markstein, P., IA-64 and Elementary Functions. Speed and Precision, Hewlett-Packard Professional Books (2000), Prentice-Hall PTR: Prentice-Hall PTR Englewood Cliffs, NJ |
| [28] | McCracken, D. D.; Dorn, W. S., Numerical Methods and Fortran Programming: With Applications in Science and Engineering (1964), Wiley: Wiley New York · Zbl 0139.12403 |
| [29] | Pichat, M., Correction d’une somme en arithmétique à virgule flottante, Numer. Math., 19, 400-406 (1972) · Zbl 0248.65027 |
| [30] | M. Pichat, Contribution à l’étude des erreurs d’arrondi en arithmétique à virgule flottante, Thèse de Doctorat d’Etat, Université de Grenoble 1, Grenoble, France, 1976 (in French).; M. Pichat, Contribution à l’étude des erreurs d’arrondi en arithmétique à virgule flottante, Thèse de Doctorat d’Etat, Université de Grenoble 1, Grenoble, France, 1976 (in French). |
| [31] | D.M. Priest, On properties of floating point arithmetics: numerical stability and the cost of accurate computations, Ph.D. Thesis, Mathematics Department, University of California, Berkeley, CA, USA, November 1992. URL:; D.M. Priest, On properties of floating point arithmetics: numerical stability and the cost of accurate computations, Ph.D. Thesis, Mathematics Department, University of California, Berkeley, CA, USA, November 1992. URL: |
| [32] | Rall, L. B., Convergence of the Newton process to multiple solutions, Numer. Math., 9, 25-37 (1966) · Zbl 0163.38702 |
| [33] | Robertazzi, T. G.; Schwartz, S. C., Best “ordering” for floating-point addition, ACM Trans. Math. Software, 14, 1, 101-110 (1988) · Zbl 0647.65033 |
| [34] | Soderquist, P.; Leeser, M., Division and square rootchoosing the right implementation, IEEE Micro, 17, 4, 56-66 (1997) |
| [35] | Stoer, J.; Bulirsch, R., Introduction to Numerical Analysis (1993), Springer: Springer New York · Zbl 0771.65002 |
| [36] | Vignes, J., A stochastic arithmetic for reliable scientific computation, Math. Comput. Simulation, 35, 233-261 (1993) |
| [37] | Wilkinson, J. H., Error analysis revisited, IMA Bull., 22, 11/12, 192-200 (1986) · Zbl 0628.65035 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
