Series Solutions of Algebraic and Differential Equations: A Comparison of Linear and Quadratic Algebraic Convergence

Abstract

Speed of convergence of Newton-like iterations in an algebraic domain can be affected heavily by the increasing cost of each step, so much so that a quadratically convergent algorithm with complex steps may be comparable to a slower one with simple steps. This note gives two examples: solving algebraic and first-order ordinary differential equations using the MACSYMA algebraic manipulation system, demonstrating this phenomenon. The relevant programs are exhibited in the hope that they might give rise to more widespread application of these techniques. 1 Newton Iteration in a Power Series Domain Newton iteration is a powerful tool for developing approximations to solutions of various types of equations. Recently, the case of iteration in a power series domain has been studied in some detail as a notion relevant especially in the field of computer-aided algebraic computation. It has been shown by Kung and Traub [6] that there are fast procedures for finding Taylor-series type expansion..