Newton’s method and high-order algorithms for the nth root computation

Two modifications of Newton’s method to accelerate the convergence of the n th root computation of a strictly positive real number are revisited. Both modifications lead to methods with prefixed order of convergence p ∈ N , p ≥ 2 . We consider affine combinations of the two modified p th-order methods which lead to a family of methods of order p with arbitrarily small asymptotic constants. Moreover the methods are of order p + 1 for some specific values of a parameter. Then we consider affine combinations of the three methods of order p + 1 to get methods of order p + 1 again with arbitrarily small asymptotic constants. The methods can be of order p + 2 with arbitrarily small asymptotic constants, and also of order p + 3 for some specific values of the parameters of the affine combination. It is shown that infinitely many p th-order methods exist for the n th root computation of a strictly positive real number for any p ≥ 3 .