Accurate floating-point summation part I: Faithful rounding

SIAM Journal on Scientific Computing

Volumn 31, Issue 1, 2008, Pages 189-224

  Rump, Siegfried M ^a,b   Ogita, Takeshi ^b,c   Oishi, Shin'ichi ^b  

^a HAMBURG UNIVERSITY OF TECHNOLOGY   (Germany)

^b WASEDA UNIVERSITY   (Japan)

^c TOKYO WOMAN'S CHRISTIAN UNIVERSITY   (Japan)

Author keywords

Distillation; Error analysis; Error free transformation; Extended and mixed precision basic linear algebra subprograms; Faithful founding; High accuracy; Maximally accurate summation; XBLAS

Indexed keywords

DIGITAL ARITHMETIC; ERROR ANALYSIS; LINEAR ALGEBRA; LINEAR TRANSFORMATIONS; NUMBER THEORY;

COMPUTING TIME; CONDITION NUMBERS; ERROR-FREE TRANSFORMATIONS; EXTENDED AND MIXED PRECISION BASIC LINEAR ALGEBRA SUBPROGRAMS; FAITHFUL FOUNDING; FLOATING POINT NUMBERS; FLOATING-POINT SUMMATION; HIGH-ACCURACY; MAXIMALLY ACCURATE SUMMATION; XBLAS;

DISTILLATION;

EID: 55049129860     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/050645671     Document Type: Article

References (50)

1. ANSI/IEEE Std 764-1985: IEEE Standard for Binary Floating-Point Arithmetic, Standards Committee of the IEEE Computer Society, New York, 1985.
2. XBLAS: A Reference Implementation for Extended and Mixed Precision BLAS, http://crd.lbl. gov/~xiaoye/XBLAS/.
3. I. J. ANDERSON, A distillation algorithm for floating-point summation, SIAM J. Sci. Comput., 20 (1999), pp. 1797-1806.
4. F. AVNAIM, J.-D. BOISSONNAT, O. DEVILLERS, F. P. PREPARATA, AND M. YVINEC, Evaluating signs of determinants using single-precision arithmetic, Algorithmica, 17 (1997), pp. 111-132.
5. D. H. BAILEY, A Fortran-90 double-double precision library, http://crd.lbl.gov/~dhbailey/mpdist/.
6. D. H. BAILEY, H. YOZO, X. S. LI, AND B. THOMPSON, ARPREC: An Arbitrary Precision Computation Package, Technical report LBNL-53651, Lawrence Berkeley National Laboratory, Berkeley, CA, 2002.
7. G. BOHLENDER., Floating-point computation of functions with maximum accuracy, IEEE Trans. Comput., C-26 (1977), pp. 621-632.
8. H. BRÖNNIMANN, C. BURNIKEL, AND S. PION, Interval arithmetic yields efficient dynamic filters for computational geometry, Discrete Appl. Math., 109 (2001), pp. 25-47.
9. H. BRÖNNIMANN AND M. YVINEC, Efficient exact evaluation of signs of determinants, Algorithmica, 27 (2000), pp. 21-56.
10. K. L. CLARKSON, Safe and effective determinant evaluation, in Proceedings of the 33rd Annual Symposium on Foundations of Computer Science, Pittsburgh, PA, IEEE Computer Society Press, 1992, pp. 387-395.
11. M. DAUMAS AND D. W. MATULA, Validated roundings of dot products by sticky accumulation, IEEE Trans. Comput., 46 (1997), pp. 623-629.
12. T. J. DEKKER, A floating-point technique for extending the available precision, Numer. Math., 18 (1971), pp. 224-242.
13. J. DEMMEL AND Y. HIDA, Accurate and efficient floating point summation, SIAM J. Sci. Comput., 25 (2003), pp. 1214-1248.
14. J. DEMMEL AND Y. HIDA, Fast and accurate floating point summation with application to computational geometry, Numer. Algorithms, 37 (2004), pp. 101-112.
15. S. GRAILLAT, P. LANGLOIS, AND N. LOUVET, Compensated Homer Scheme, in Algebraic and Numerical Algorithms and Computer-Assigned Proofs, B. Buchberger, S. Oishi, M. Plum, and S. M. Rump, eds., Dagstuhl Seminar Proceedings, 2006.
16. J. R. HAUSER, Handling floating-point exceptions in numeric programs, ACM Trans. Program. Lang. Syst., 18 (1996), pp. 139-174.
17. C. HECKER, Let's get to the (floating) point, Game Developer, 2 (1996), pp. 19-24.
18. N. J. HIGHAM, The accuracy of floating point summation, SIAM J. Sci. Comput., 14 (1993), pp. 783-799.
19. N. J. HIGHAM, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, 2002.
20. C. M. HOFFMANN, Robustness in geometric computations, J. Comput. Inform. Sci. Eng., l (2001), pp. 143-156.
21. M. JANKOWSKI, A. SMOKTUNOWICZ, AND H. WOŹNIAKOWSKI, A note on floating-point summation of very many terms, Electron. Informationsverarb. Kybernet., 19 (1983), pp. 435-440.
22. M. JANKOWSKI AND H. WOŹNIAKOWSKI, The accurate solution of certain continuous problems using only sigle precision arithmetic, BIT, 25 (1985), pp. 635-651.
23. W. KAHAN, A survey of error analysis, in Proceedings of the IFIP Congress, Information Processing 71, North-Holland, Amsterdam, 1972, pp. 1214-1239.
24. W. KAHAN, Implementation of Algorithms (Lecture Notes by W. S. Haugeland and D. Hough), Technical report 20, Department of Computer Science, University of California, Berkeley, CA, 1973.
25. A. KIEŁBASZIŃSKI, Summation algorithm with corrections and some of its applications, Math. Stos, 1 (1973), pp. 22-41.
26. D. E. KNUTH, The Art of Computer Programming: Seminumerical Algorithms, Vol. 2, Addison-Wesley, Reading, MA, 1969.
27. S. KRISHNAN, M. FOSKEY, T. CULVER, J. KEYSER, AND D. MANOCHA, PRECISE: Efficient multiprecision evaluation of algebraic roots and predicates for reliable geometric computation, in Proceedings of the 17th Annual Symposium on Computational Geometry, New York, ACM Press, 2001, pp. 274-283.
28. U. KULISCH AND W. L. MIRANKER, Arithmetic operations in interval spaces, Comput., Suppl. 2 (1980), pp. 51-67.
29. P. LANGLOIS, Accurate Algorithms in Floating Point Arithmetic, in an invited talk at the 12th GAMM-IMACS International Symposion on Scientific Computing, Computer Arithmetic and Validated Numerics, Duisburg, 2006.
30. P. LANGLOIS AND N. LOUVET, Solving Triangular Systems More Accurately and Efficiently, in Proceedings of the 17th IMACS World Congress, Paris, 2005.
31. H. LEUPRECHT AND W. OBERAIGNER, Parallel algorithms for the rounding exact summation of floating point numbers, Computing, 28 (1982), pp. 89-104.
32. 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. YOO, Design, implementation and testing of extended and mixed precision BLAS, ACM Trans. Math. Software, 28 (2002), pp. 152-205.
33. S. LINNAINMAA, Software for doubled-precision floating point computations, ACM Trans. Math. Software, 7 (1981), pp. 272-283.
34. M. MALCOLM, On accurate floating-point summation, Comm. ACM, 14 (1971), pp. 731-736.
35. O. MØLLER, Quasi double precision in floating-point arithmetic, BIT, 5 (1965), pp. 37-50.
36. A. NEUMAIER, Rundungsfehleranalyse einiger Verfahren zur Summation endlicher Summen, Z. Angew. Math. Mech., 54 (1974), pp. 39-51.
37. T. OGITA, S. M. RUMP, AND S. OISHI, Accurate sum and dot product, SIAM J. Sci. Comput., 26 (2005), pp. 1955-1988.
38. K. OZAKI, T. OGITA, S. M. RUMP, AND S. OISHI, Fast and robust algorithm for geometric predicates using floating-point arithmetic, Trans. Japan Soc. Ind. Appl. Math., 16 (2006), pp. 553-562.
39. M. PICHAT, Correction d'une somme en arithmétique à virgule flottante, Numer. Math., 19 (1972), pp. 400-406.
40. D. M. PRIEST, Algorithms for arbitrary precision floating point arithmetic, in Proceedings of the 10th Symposium on Computer Arithmetic, P. Kornerup and D. W. Matula, eds., IEEE Press, Grenoble, 1991, pp. 132-145.
41. 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, 1992.
42. D. R. ROSS, Reducing truncation errors using cascading accumulators, Comm. ACM, 8 (1965), pp. 32-33.
43. S. M. RUMP, INTLAB - INTerval LABoratory, in Developments in Reliable Computing, T. Csendes, ed., Kluwer Academic Publishers, Dordrecht, 1999, pp. 77-104.
44. S. M. RUMP, T. OGITA, AND S. OISHI, Accurate Floating-point Summation Part II: Sign, K-fold Faithful and Rounding to Nearest, SIAM J. Sci. Comput., submitted; also available online from http://www.ti3.tu-harburg.de/rump.
45. S. SCHIRRA, Precision and robustness in geometric computations, in Algorithmic Foundations of Geographic Information Systems, Lect. Notes Comput. Sci. 1340, M. van Kreveld, J. Nievergelt, T. Roos, and P. Widmayer, eds., Springer-Verlag, Berlin, 1997, pp. 255-287.
46. J. R. SHEWCHUK, Adaptive precision floating-point arithmetic and fast robust geometric predicates, Discrete Comput. Geom., 18 (1997), pp. 305-363.
47. J. M. WOLFE, Reducing truncation errors by programming, Comm. ACM, 7 (1964), pp. 355-356.
48. Y.-K. ZHU AND W. HAVES, Fast, guaranteed-accurate sums of many floating-point numbers, in Proceedings of the 7th Conference on Real Numbers and Computers, G. Hanrot and P. Zimmermann, eds., 2006, pp. 11-22.
49. Y.-K. ZHU, J.-H. YONG, AND G.-Q. ZHENG, A new distillation algorithm for floating-point summation, SIAM J. Sci. Comput., 26 (2005), pp. 2066-2078.
50. G. ZIELKE AND V. DRYGALLA, Genaue Lösung Linearer Gleichungssysteme, GAMM Mitt. Ges. Angew. Math. Mech., 26 (2003), pp.7-107.