Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

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
Full-text PDF Free Access

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.


References [Enhancements On Off] (What's this?)

  • [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.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65.50

Retrieve articles in all journals with MSC: 65.50


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