Choosing starting values for certain Newton-Raphson iterations

Theoretical Computer Science

Volumn 351, Issue 1, 2006, Pages 101-110

  Kornerup, Peter ^a   Muller, Jean Michel ^b  

^a UNIVERSITY OF SOUTHERN DENMARK   (Denmark)

^b CNRS   (France)

Author keywords

Computer arithmetic; Division; Newton Raphson iteration; Root extraction; Square root; Square root reciprocal

Indexed keywords

APPROXIMATION THEORY; COMPUTER SCIENCE; DIGITAL ARITHMETIC; FUNCTION EVALUATION; ITERATIVE METHODS;

COMPUTER ARITHMETIC; DIVISION; NEWTON-RAPHSON ITERATION; ROOT EXTRACTION; SQUARE-ROOT; SQUARE-ROOT RECIPROCAL;

NUMBER THEORY;

EID: 31144443150     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.tcs.2005.09.056     Document Type: Conference Paper

References (19)

1. W.J. Cody Double precision square root for the CDC-3600 Comm. ACM 7 12 1964 715 718
2. A. Dahan-Dalmedico, J. Peiffer, Histoire des Mathématiques, Editions du Seuil, Paris, 1986 (in French).
3. D. DasSarma, and D.W. Matula Measuring the accuracy of ROM reciprocal tables IEEE Trans. Comput. 43 8 1994 932 940
4. J. Eve Starting approximations for the iterative calculation of square roots Comput. J. 6 1963 274 276
5. C.T. Fike Starting approximations for square root calculation on IBM system/360 Comm. ACM 9 4 1966 297 299
6. D. Fowler, and E. Robson Square root approximations in old Babylonian mathematics: YBC 7289 in context Historia Math. 25 1998 366 378
7. H.J. Maehly, Approximations for the CDC 1604, Technical Report, Control Data Corp., 1960.
8. P.W. Markstein Computation of elementary functions on the IBM RISC System/6000 Processor IBM J. Res. Develop. 34 1 1990 111 119
9. P.W. Markstein IA-64 and Elementary Functions: Speed and Precision, Hewlett-Packard Professional Books 2000 Prentice-Hall Englewood Cliffs, NJ ISBN: 0130183482
10. P. Montuschi, and M. Mezzalama Optimal absolute error starting values for Newton-Raphson calculation of square root Computing 46 1991 67 86
11. D.G. Moursund Optimal starting values for Newton-Raphson calculation of x Comm. ACM 10 7 1967 430 432
12. I. Newton, Methodus Fluxionem et Serierum Infinitarum, 1664-1671.
13. S.F. Oberman, Floating-point division and square root algorithms and implementation in the AMD-k7 microprocessor, in: I. Koren, P. Kornerup (Eds.), Proc. 14th IEEE Symp. Computer Arithmetic (Adelaide, Australia), Los Alamitos, CA, April 1999, IEEE Computer Society Press, Silverspring, MD, pp. 106-115.
14. C.V. Ramamoorthy, J.R. Goodman, and K.H. Kim Some properties of iterative square-rooting methods using high-speed multiplication IEEE Trans. Comput. C-21 1972 837 847 Reprinted in E.E. Swartzlander, Computer Arithmetic, Vol. 1, IEEE Computer Society Press Tutorial, Los Alamitos, CA, 1990
15. D. Russinoff A mechanically checked proof of IEEE compliance of a register-transfer-level specification of the AMD-k7 floating-point multiplication, division, and square root instructions LMS J. Comput. Math. 1 1998 148 200
16. M.J. Schulte, J. Omar, and E.E. Swartzlander Optimal initial approximation for the Newton-Raphson division algorithm Computing 53 1994 233 242
17. P. Sebah, X. Gourdon, Newton's method and high order iterations, Technical Report, 2001. .
18. P.H. Sterbenz, and C.T. Fike Optimal starting approximations for Newtons method Math. Comp. 23 1969 313 318
19. M. Wayne Wilson Optimal starting approximations for generating square root for slow or no divide Comm. ACM 13 9 1970 559 560