On Local Roundoff Errors in Floating-Point Arithmetic

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On Local Roundoff Errors in Floating-Point Arithmetic

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Journal of the ACM (JACM) archive
Volume 20 , Issue 3 (July 1973) table of contents

Pages: 391 - 398

Year of Publication: 1973

ISSN:0004-5411

Authors

Toyohisa Kaneko	IBM Thomas J Watson Research Center, Yorktown Heights, NY and Princeton University, Princeton, New Jersey
Bede Liu	Department of Electrical Engineering, Brackett Hall, Engineering Quadrangle, Princeton University, Princeton, New Jersey

Publisher

ACM New York, NY, USA

Bibliometrics

Downloads (6 Weeks): 8, Downloads (12 Months): 53, Citation Count: 7

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ABSTRACT

A bound on the relative error in floating-point addition using a single-precision accumulator with guard digits is derived. It is shown that even with a single guard digit, the accuracy can be almost as good as that using a double-precision accumulator. A statistical model for the roundoff error in double-precision multiplication and addition is also derived. The model is confirmed by experimental measurements.

REFERENCES

Note: OCR errors may be found in this Reference List extracted from the full text article. ACM has opted to expose the complete List rather than only correct and linked references.

	1	BRUNT, I~ICNAI~D P On the best choice o~ a base for flouting-point number representation (submitted for publication).
	2	Floatlng-po~nt arithmetic In IBM System/360 Principles of Operation, IBM Systems Reference Ltbrary, Form A22-6821-7 (Sept. 1968), pp 41-50 3
	3	FORSYTHE, G , AND MOLhR, C B Computer Solutwt~. of L~t~ear Algebra, c Systems Prentice- Hall, Englewood Cliffs, N J, 1967.
	4	HAMMING, R.W. On the dmtrlbutlon of numbers. Bell Syst. Tech. J. 49, 8 (Oct. 1970), 1609-1626.
	5	HENRICI, P D~screte Vamable Methods ~ Ord~ary D~fferent~al Equatwns Wiley, New York, 1968.
	6	Toyohisa Kaneko , Bede Liu, Accumulation of Round-Off Error in Fast Fourier Transforms, Journal of the ACM (JACM), v.17 n.4, p.637-654, Oct. 1970 [doi>10.1145/321607.321613]
	7	H. Kuki , W. J. Cody, A statistical study of the accuracy of floating point number systems, Communications of the ACM, v.16 n.4, p.223-230, April 1973 [doi>10.1145/362003.362013]
	8	LIu, B, AND KANWKO, T Error analysis of digital filters realized with floating-point arithmetic. Proc IEEE 57, 10 (Oct 1969), 1735-1747.
	9	SWEENEY, D. W An analysis of floatlng-point addition. IBM Syst J ~4, 1 (1965), 31-42
	10	WIDROW, B. Statistical analyms of amplitude-quantized sampled-data systems, AIEE Trans. 59 (Jan. 1961), 555-568.
	11	WILmNSON, J. H Roundzng Errors in Algebraic Processes. Prentice-Hall, Englewood Cliffs, N. J, 1963.

CITED BY 7

	Alan Feldstein , Richard Goodman, Convergence Estimates for the Distribution of Trailing Digits, Journal of the ACM (JACM), v.23 n.2, p.287-297, April 1976
	Tran-Thong , Bede Liu, Floating Point Fast Fourier Transform Computation Using Double Precision Floating Point Accumulators, ACM Transactions on Mathematical Software (TOMS), v.3 n.1, p.54-59, March 1977
	Joaquín Bustoz , Alan Feldstein , Richard Goodman , Seppo Linnainmaa, Improved Trailing Digits Estimates Applied to Optimal Computer Arithmetic, Journal of the ACM (JACM), v.26 n.4, p.716-730, Oct. 1979
	H. Kuki , W. J. Cody, A statistical study of the accuracy of floating point number systems, Communications of the ACM, v.16 n.4, p.223-230, April 1973
	H. Kuki , W. J. Cody, A statistical study of the accuracy of floating point number systems, Communications of the ACM, v.26 n.1, p.79-83, Jan. 1983
	Nai-Kuan Tsao, On the distributions of significant digits and roundoff errors, Communications of the ACM, v.17 n.5, p.269-271, May 1974
	W. J. Cody, Software basics for computational mathematics, ACM SIGNUM Newsletter, v.15 n.2, p.18-29, June 1980