Best rational starting approximations and improved Newton iteration for the square root

Author:
Ichizo Ninomiya

Journal:
Math. Comp. **24** (1970), 391-404

MSC:
Primary 65.50

DOI:
https://doi.org/10.1090/S0025-5718-1970-0273809-4

MathSciNet review:
0273809

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Abstract | References | Similar Articles | Additional Information

Abstract: The most important class of the best rational approximations to the square root is obtained analytically by means of elliptic function theory. An improvement of the Newton iteration procedure is proposed.

**[1]**N. I. Achieser,*Theory of approximation*, Translated by Charles J. Hyman, Frederick Ungar Publishing Co., New York, 1956. MR**0095369****[2]**P. L. Chebyshev, "Sur les expressions approchées de la racine carrée d'une variable par des fractions simples,"*Oeuvres*, Vol. 2, Chelsea, New York, pp. 542-558.**[3]**P. L. Chebyshev, "Sur les fractions algébriques qui représentent approximativement la racine carrée d'une variable comprise entre les limites données," Oeuvres, Vol. 2, Chelsea, New York, p. 725.**[4]**W. J. Cody, "Double-precision square root for the CDC-3600,"*Comm. ACM*, v. 7, 1964, pp. 715- 718.**[5]**J. Eve, "Starting approximations for the iterative calculation of square roots,"*Comput. J.*, v. 6, 1963, pp. 274-276.**[6]**C. T. Fike, "Starting approximations for square root calculation on IBM system/360,"*Comm. ACM*, v. 9, 1966, pp. 297-299.**[7]**David G. Moursund,*Optimal starting values for Newton-Raphson calculation of √𝑥*, Comm. ACM**10**(1967), 430–432. MR**0240952**, https://doi.org/10.1145/363427.363454**[8]**Richard F. King and David L. Phillips,*The logarithmic error and Newton’s method for the square root*, Comm. ACM**12**(1969), 87–88. MR**0285109**, https://doi.org/10.1145/362848.362861**[9]**P. H. Sterbenz & C. T. Fike, "Optimal starting approximations for Newton's method,"*Math. Comp.*, v. 23, 1969, pp. 313-318.**[10]**Ichizo Ninomiya,*Generalized rational Chebyshev approximation*, Math. Comp.**24**(1970), 159–169. MR**0261229**, https://doi.org/10.1090/S0025-5718-1970-0261229-8**[11]**H. E. Salzer, "Quick calculation of Jacobian elliptic functions,"*Comm. ACM*, v. 5, 1962, p. 399.**[12]**Milton Abramowitz and Irene A. Stegun,*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642****[13]**J. Tannery & J. Molk,*Éléments de la Théorie des Fonctions Elliptiques*, Vol. 1, Gauthier-Villars. Paris, 1893, pp. 294-295.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0273809-4

Keywords:
Rational approximation,
best approximation,
square root,
Newton iteration,
starting approximation,
Chebyshev's criterion,
Moursund's criterion,
logarithmic criterion,
Jacobian elliptic function,
transformation of elliptic functions

Article copyright:
© Copyright 1970
American Mathematical Society