High order iterative methods for approximating square roots

Abstract

Given any natural numberm ≥ 2, we describe an iteration functiong m (x) having the property that for any initial iterate\(x_0 > \sqrt \alpha \), the sequence of fixed-point iterationx k +1 =g m (x k ) converges monotonically to\(\sqrt \alpha \) having anm-th order rate of convergence. Form = 2 and 3,g m (x) coincides with Newton's and Halley's iteration functions, respectively, as applied top(x) =x2 − α.