Numerical Analysis in Lie Groups

There is growing recognition in the last few years that Lie groups and homogeneous spaces are often the right connguration space for the dis-cretization of time-dependent diierential equations. In this paper we review brieey recent advances in Lie-group calculations, concentrating mainly on approximation methods that advance a trivialised version of the diierential equation in a Lie algebra in terms of either Magnus or Cayley expansions. 1 Why Lie groups? The subject matter of this paper is as old as computational mathematics itself, yet still replete with exciting challenges: the numerical solution of a time-dependent ordinary diierential system y 0 = f(t; y); t t 0 ; y(t 0) = y 0 ; (1.1) where f 2 Lip((t 0 ; 1)R m ! R m). Practical methods for the equation (1.1), e.g. Runge{Kutta, multistep and extrapolation schemes, are well known to all students of numerical analysis and they are accompanied by powerful analysis and quality software. Our rst goal in this paper is thus to motivate the consideration of an entirely new breed of time-stepping algorithms, of a form unfamiliar to most practitioners and, indeed, based upon a radically diierent approach to the entire subject area. Important qualitative and structural features of (1.1) can be often phrased in the language of diierential geometry, an issue discussed at greater length in 3]. In particular, it is often known that the conngura-tion space of the solution is a diierentiable manifold, y(t) 2 M R m ,