Optimal starting values for Newton-Raphson calculation of x1 2

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Optimal starting values for Newton-Raphson calculation of x^{1 2}

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Communications of the ACM archive
Volume 10 , Issue 7 (July 1967) table of contents

Pages: 430 - 432

Year of Publication: 1967

ISSN:0001-0782

Author

David G. Moursund

Michigan State Univ., East Lansing

Publisher

ACM New York, NY, USA

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Downloads (6 Weeks): 4, Downloads (12 Months): 23, Citation Count: 9

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ABSTRACT

The problem of obtaining starting values for the Newton-Raphson calculation of √x on a digital computer is considered. It is shown that the conventionally used best uniform approximations to √x do not provide optimal starting values. The problem of obtaining optimal starting values is stated, and several basic results are proved. A table of optimal polynomial starting values is given.

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	EVE, J. Starting approximations for the iterative calculation of square roots. Comput. J. 6 (Oct. 1963), 274-276.
	2	C. T. Fike, Starting approximations for square root calculation on IBM system /360, Communications of the ACM, v.9 n.4, p.297-299, April 1966 [doi>10.1145/365278.365556]
	3	MOURSUND, D. G. Chebyshev approximation using a generalized weight function. SIAM J. Numer. Anal. 8, 3 (Sept. 1966), 435-450.
	4	MOURSUND, D. G. Computational aspects of Chebyshev approximation using a generalized weight function. (In preparation)

CITED BY 9

	Richard F. King , David L. Phillips, The logarithmic error and Newton's method for the square root, Communications of the ACM, v.12 n.2, p.87-88, Feb. 1969
	M. Wayne Wilson, Optimal starting approximations for generating square root for slow or no divide, Communications of the ACM, v.13 n.9, p.559-560, Sept. 1970
	Nabil Rousan , Hani Abu-Salem, Computation of √x, Proceedings of the 1999 ACM symposium on Applied computing, p.74-77, February 28-March 02, 1999, San Antonio, Texas, United States
	Artenio De Matteis, Improvements based on computed errors, Communications of the ACM, v.10 n.11, p.683, Nov. 1967
	G. A. Watson, On an algorithm for nonlinear minimax approximation, Communications of the ACM, v.13 n.3, p.160-162, March 1970
	M. Andrews , S. F. McCormick , G. D. Taylor, Evaluation of the square root function on microprocessors, Proceedings of the annual conference, p.185-191, October 20-22, 1976, Houston, Texas, United States
	Peter Kornerup , Jean-Michel Muller, Choosing starting values for certain Newton-Raphson iterations, Theoretical Computer Science, v.351 n.1, p.101-110, 14 February 2006
	C. T. Fike, A rational approximation optimal by Moursund's criterion, Communications of the ACM, v.10 n.11, p.683-684, Nov. 1967
	E. M. Reingold, Establishing lower bounds on algorithms: a survey, Proceedings of the November 16-18, 1971, fall joint computer conference, November 16-18, 1971, Las Vegas, Nevada