Accurate and efficient floating point summation

SIAM Journal on Scientific Computing

Volumn 25, Issue 4, 2003, Pages 1214-1248

  Demmel, James ^a   Hida, Yozo ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Accurate summation; Distillation; Dot product; Floating point; Roundoff error

Indexed keywords

ALGORITHMS; ERROR ANALYSIS; SORTING; THEOREM PROVING; VECTORS;

ACCURATE SUMMATION; DOT PRODUCTS; FLOATING POINT; ROUNDOFF ERRORS;

DIGITAL ARITHMETIC;

EID: 4043118275     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827502407627     Document Type: Article

References (25)

1. ANSI/IEEE, IEEE Standard for Binary Floating Point Arithmetic, Std 754-1985 edition, IEEE, New York, 1985.
2. ANSI/IEEE, IEEE Standard for Radix Independent Floating Point Arithmetic, Std 854-1987 edition, IEEE, New York, 1987.
3. G. BOHLENDER, Floating point computation of functions with maximum accuracy, IEEE Trans. Comput., C-26 (1977), pp. 621-632.
4. T. DEKKER, A floating point technique for extending the available precision, Numer. Math., 18 (1971), pp. 224-242.
5. J. DEMMEL AND Y. HIDA, Fast and accurate floating point summation with application to computational geometry, in Proceedings of the 10th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2002), submitted.
6. N. J. HIGHAM, The accuracy of floating point summation, SIAM J. Sci. Comput., 14 (1993), pp. 783-799.
7. N. J. HIGHAM, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, 2002.
8. IEEE, Proposed Revision of the IEEE Floating Point Arithmetic Standard 754, IEEE, New York, 2001. http://grouper.ieee.org/groups/754.
9. INTEL CORPORATION, Intel Itanium Architecture Software Developer's Manual, Volume 1, Intel Corporation, Santa Clara, CA, 2002. http://developer. intel.com/design/itanium/manuals.htm.
10. INTEL CORPORATION, Intel Architecture Software Developer's Manual, Volume 1, Basic Architecture, Intel Corporation, Santa Clara, CA, 2002. http://developer.intel.com/design/ pentium/manuals.htm.
11. W. KAHAN, Doubled-precision IEEE Standard 754 Floating Point Arithmetic, manuscript, 1987.
12. D. KNUTH, The Art of Computer Programming, Volume 2, Addison-Wesley, Reading, MA, 1969.
13. U. KULISCH AND G. BOHLENDER, Formalization and implementation of floating-point matrix operations, Computing, 16 (1976), pp. 239-261.
14. U. KULISCH AND W. L. MIRANKER, Computer Arithmetic in Theory and Practice, Academic Press, New York, 1981
15. H. LEUPRECHT AND W. OBERAIGNER, Parallel algorithms for the rounding exact summation of floating point numbers, Computing, 28 (1982), pp. 89-104.
16. S. LINNAINMAA, Software for doubled-precision floating point computations, ACM Trans. Math. Software, 7 (1981), pp. 272-283.
17. M. MALCOLM, On accurate floating-point summation, Comm. ACM, 14 (1971), pp. 731-736.
18. O. MØLLER, Quasi double precision in floating-point arithmetic, BIT, 5 (1965), pp. 37-50.
19. M. PICHAT, Correction d'une somme en arithmetique à virgule flottante, Numer. Math., 19 (1972), pp. 400-406.
20. D. PRIEST, Algorithms for arbitrary precision floating point arithmetic, in Proceedings of the 10th Symposium on Computer Arithmetic, Grenoble, France, June 26-28, 1991, P. Kornerup and D. Matula, eds., IEEE, New York, pp. 132-145.
21. D. PRIEST, On Properties of Floating Point Arithmetics: Numerical Stability and the Cost of Accurate Computations, Ph.D. thesis, University of California at Berkeley, 1992. ftp.icsi. berkeley.edu/pub/theory/priest-thesis. ps.Z.
22. D. R. Ross, Reducing truncation errors using cascading accumulators, Comm. ACM, 8 (1965), pp. 32-33.
23. C. P. SCHNORH AND M. EUCHNER, Lattice basis reduction: Improved practical algorithms and solving subset sum problems, Math. Programming, 66 (1994), pp. 181-199.
24. J. R. SHEWCHUK, Adaptive precision floating-point arithmetic and fast robust geometric predicates, Discrete Comput. Geom., 18 (1997), pp. 305-363.
25. J. M. WOLFE, Reducing truncation errors by programming, Comm. ACM, 7 (1964), pp. 355-356.