Series Solutions of Algebraic and Diierential Equations: a Comparison of Linear and Quadratic Algebraic Convergence 1 Newton Iteration in a Power Series Domain

Speed of convergence of Newton-like iterations in an algebraic domain can be aaected 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 rst-order ordinary diierential 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. 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 eld of computer-aided algebraic computation. It has been shown by Kung and Traub 6] that there are fast procedures for nding Taylor-series type expansions for any algebraic function. Based on a reduction of an arbitrary algebraic problem to a \regular" problem , their paper then uses a Newton iteration to solve the problem \fast." The theorem below provides a constructive procedure for computing the expansion of a regular problem. In the next section we describe how to extend this type of iteration to rst order diierential equations. 0 One of our interests here is to see if the asymptot-ically \fast" algorithms are, in practice, faster than more naive ones, especially on simple inputs. A second motivation for this paper is to provide simple implementations of linearly and quadratically convergent iterations in a popular algebraic manipulation system (MACSYMA). The theorems we state have been derived and proved elsewhere, and are quoted for reference. We quote from a reference ((7]), Theorem 1 Let f(x) be a polynomial with coeecients in a power series domain D = Ft]]. By this we denote the domain of truncated power series in t with coee-cients in F. Let in F be an O(t) approximation to a root of f(x). (i.e. x = is a solution to f(x) = 0 when t = 0). Furthermore, suppose that satisses f 0 () 6 = 0 when t = 0 and where the prime indicates diierentia-tion with respect to x.