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

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