Efficient computation of truncated power series: Direct approach versus newton's method - Koepf, Wolfram, (Utilitas Mathematica, 2010)
Newton's method is used to approximate the zeros of a real function f(x). In the generic case, Newton's method is quadratically convergent, i. e. in each step the number of correct decimal digits roughly doubles. It is also wellknown that Newton's method can be similarly used to compute the Taylor coefficients of a function x(t), given implicitly by f(t, x(t)) = 0, in an iterative way such that in each iteration step the number of correct coefficients doubles. In this paper, we present implementations of higher order iteration schemes generalizing Newton's method which enable us to triple, quadruple, quintuple etc. the number of correct Taylor coefficients of an implicitly given function in each iteration step. These algorithms are of theoretical interest, but in practice they turn out to be not as efficient as the "usual" Newton method since the expression swell generated by the complexity of the formulas is higher than the advantage by the higher order of the method. We give examples in Mathematica and in Maple showing that in certain cases Newton's (implicit) method is faster than the direct (explicit) computation.