TITLE:
Newton, Halley, Pell and the Optimal Iterative High-Order Rational Approximation of √N
AUTHORS:
Isaac Fried
KEYWORDS:
Iterative Methods, Super-Linear and Super-Quadratic Methods, Square Roots, Pell’s Equation, Optimal Rational Iterants, Root Bounds
JOURNAL NAME:
Applied Mathematics,
Vol.9 No.7,
July
30,
2018
ABSTRACT: In this paper we examine single-step iterative methods for the solution of the
nonlinear algebraic equation f (x) = x2 - N = 0 , for some integer N, generating
rational approximations p/q that are optimal in the sense of Pell’s equation p2 - Nq2 = k for some integer k, converging either alternatingly or
oppositely.