Optimal partitioning of Newton’s method for calculating roots

Abstract

In this paper, an algorithm is given for calculating roots via Newton’s method initialized with a piecewise best starting approximation. The piecewise best starting approximation corresponds to an optimal partitioning of the interval of the domain of Newton’s method. Explicit formulas are given when piecewise linear polynomials are used for the best starting approximations. Specific tables are given for square roots, cube roots and reciprocal square roots.