Newton's method: A great algebraic algorithm

Abstract

The analytic concepts of approximation, convergence, differentiation, and Taylor series expansion are applied and interpreted in the context of an abstract power series domain. Newton's method is then shown to be applicable to solving for a power series root of a polynomial with power series coefficients, resulting in fast algorithms for a variety of power series manipulation problems. Sample applications of a FORMAC implementation of Newton's method as an algebraic algorithm are presented.