Research ArticleOpen AccessGoogle Scholar indexed
Newton, Halley, Pell and the Optimal Iterative High-Order Rational Approximation of √<span style='margin-left:-2px;margin-right:2px; border-top:1px solid black'>N</span>
Department of Mathematics, Boston University, Boston, MA, USA
- 1 Department of Mathematics, Boston University, Boston, MA, USA
Copy link · social · email
Abstract
In this paper we examine single-step iterative methods for the solution of the nonlinear algebraic equation f ( x ) = x 2 - N = 0 , for some integer N , generating rational approximations p / q that are optimal in the sense of Pell’s equation p 2 - Nq 2 = k for some integer k , converging either alternatingly or oppositely.
KeywordsIterative MethodsSuper-Linear and Super-Quadratic MethodsSquare RootsPell’s EquationOptimal Rational IterantsRoot Bounds
- Silverman, J.H. (2006) A Friendly Introduction to Number Theory, Chapter 12. Prentice Hall, New Jersey.
- Whitford, E.E. (1912) The Pell Equation. E. E. Whitford, New York.
- Barbeau, E.J. (2000) Pell’s Equation. Springer, New York.
- Jacobson Jr., M.J. and Williams, H.C. (2009) Solving the Pell Equation. Springer, New York.
- Fried, I. (2009) Oppositely Converging Newton-Raphson Method for Nonlinear Equilibrium Problems. International Journal for Numerical Methods in Engineering, 79, 375-378. https://doi.org/10.1002/nme.2574
- Fried, I. (2013) High-Order Iterative Bracketing Methods. International Journal for Numerical Methods in Engineering, 94, 708-714. https://doi.org/10.1002/nme.4467
- Fried, I. (2014) Effective High-Order Iterative Methods via the Asymptotic Form of the Taylor-Lagrange Remainder. Journal of Applied Mathematics, 2014, Article ID: 108976. https://doi.org/10.1155/2014/108976
- Brown, L.M. (1997) An Algorithm for Square Roots: An Episode in the Campaign Against Dotage. The Mathematical Gazette, 81, 428-429. https://doi.org/10.2307/3619622
- McBride, A. (1999) Remarks on Pell’s Equation and Square Root Algorithms. The Mathematical Gazette, 83, 47-52. https://doi.org/10.2307/3618682
- Dunkel, O. (1927) A Note on the Computation of Arithmetic Roots. The American Mathematical Monthly, 34, 366-368. https://doi.org/10.2307/2300041
- Uspensky, J.V. (1927) Note on the Computation of Roots. The American Mathematical Monthly, 34, 130-134. https://doi.org/10.1080/00029890.1927.11986665
- Neta, B. (1979) A Sixth-Order Family of Methods for Nonlinear Equations. International Journal of Computer Mathematics, Section B, 7, 157-161. https://doi.org/10.1080/00207167908803166
- Fried, I. (2016) A Remarkable Chord Iterative Method for Roots of Uncertain Multiplicity. Applied Mathematics, 7, 1207-1214. https://doi.org/10.4236/am.2016.711106