Solutions of ODEs with Removable Singularities

Abstract

Automatic differentiation (AD) can be used to compute a Taylor polynomial approximation to a function x(t) of a single variable at a base point t0 to any desired degree. AD can handle functions with removable singularities, such as x(t) = t/(1 − cos t) at t0 = 0. The standard convolution algorithm for computing the expansion of a quotient H = F/G from F = G ∗ H, given the expansions of F and G, must first be preprocessed by determining the lowest terms t and t appearing in F and G, respectively. Then if m ≥ n, the algorithm, slightly modified, works. AD also can be used to compute Taylor polynomial approximations to solutions of the IVP (Initial Value Problem) for a single ODE (Ordinary Differential Equation) or a system of ODEs of the form dx dt = f(t, x), x(t0) = x0 .