Interpolation in Lie Groups

Abstract

We consider interpolation in Lie groups. Based on points on the manifold together with tangent vectors at (some of) these points, we construct Hermite interpolation polynomials. If the points and tangent vectors are produced in the process of integrating an ordinary differential equation in terms of Lie-algebra actions, we use the truncated inverse of the differential of the exponential mapping and the truncated Baker--Campbell--Hausdorff formula to relatively cheaply construct an interpolation polynomial. Much effort has lately been put into research on geometric integration, i.e., the process of integrating differential equations in such a way that the configuration space of the true solution is respected by the numerical solution. Some of these methods may be viewed as generalizations of classical methods, and we investigate the construction of intrinsic dense output devices as generalizations of the continuous Runge--Kutta methods.