High order iterative methods for approximating square roots

Given any natural numberm 2, we describe an iteration functiong_m(x) having the property that for any initial iterate, the sequence of fixed-point iterationx_k+1 =g_m(x_k) converges monotonically to 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 – .This research is supported in part by the National Science Foundation under Grant No. CCR-9208371.