In this paper we give precise definitions of different, properly invariant notions of mean or average rotation. Each mean is associated with a metric in SO(3). The metric induced from the Frobenius inner product gives rise to a mean rotation that is given by the closest special orthogonal matrix to the usual arithmetic mean of the given rotation matrices. The mean rotation associated with the intrinsic metric on SO(3) is the Riemannian center of mass of the given rotation matrices. We show that the Riemannian mean rotation shares many common features with the geometric mean of positive numbers and the geometric mean of positive Hermitian operators. We give some examples with closed-form solutions of both notions of mean.

Abstract — A numerical integrator is derived for a class of models that describe the attitude dynamics of a rigid body in the presence of an attitude dependent potential. The numerical integrator is obtained from a discrete variational principle, and exhibits excellent geometric conservation properties. In particular, by performing computations at the level of the Lie algebra, and updating the solution using the matrix exponential, the attitude automatically evolves on the rotation group embedded in the space of matrices. The geometric conservation properties of the numerical integrator imply long time numerical stability. We apply this variational integrator to the uncontrolled 3D pendulum, that is a rigid asymmetric body supported at a frictionless pivot acting under the influence of uniform gravity. Interesting dynamics of the 3D pendulum are exposed. I.

Commencing from a global-error formula, originally due to Henrici, we investigate the accumulation of global error in the numerical solution of linear highly-oscillating systems of the form y 00 + g(t)y = 0, where g(t) t!1 \Gamma! 1. Using WKB analysis we derive an explicit form of the global-error envelope for Runge-Kutta and Magnus methods. Our results are closely matched by numerical experiments. Motivated by the superior performance of Lie-group methods, we present a modification of the Magnus expansion which displays even better long-term behaviour in the presence of oscillations.

Numerical methods based on the Magnus expansion are an ecient class of integrators for Schrodinger equations with time-dependent Hamiltonian. Though their derivation assumes an unreasonably small time step size as would be required for a standard explicit integrator, the methods perform well even for much larger step sizes. This favorable behavior is explained, and optimal-order error bounds are derived which require no or only mild restrictions of the step size. In contrast to standard integrators, the error does not depend on higher time derivatives of the solution, which is in general highly oscillatory.

Abstract. Variational techniques are used to analyze the problem of rigid-body dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions, and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem. Discretizations of this nonsmooth mechanics are developed by using the methodology of variational discrete mechanics. This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the long-time stable energy behavior typical of variational methods is demonstrated.

Abstract In this paper, structure-preserving time-integrators for rigid body-type mechanical

In this paper, Smale’s α theory is generalized to the context of intrinsic Newton iteration on geodesically complete analytic Riemannian and Hermitian manifolds. Results are valid for analytic mappings from a manifold to a linear space of the same dimension, or for analytic vector fields on the manifold. The invariant γ is defined by means of high order covariant derivatives. Bounds on the size of the basin of quadratic convergence are given. If the ambient manifold has negative sectional curvature, those bounds depend on the curvature. A criterion of quadratic convergence for Newton iteration from the information available at a point is also given. 1 Introduction and main results. Numerical problems posed in manifolds arise in many natural contexts. Classical examples are given by the eigenvalue problem, the symmetric eigenvalue problem, invariant subspace computations, minimization problems with orthogonality constraints, optimization problems with equality constraints... etc. In the first

In this paper we describe the use of the theory of generalized polar decompositions (MuntheKaas, Quispel & Zanna 2000b) to approximate a matrix exponential. The algorithms presented in this paper have the property that, if Z 2 g, a Lie algebra of matrices, then the approximation for exp(Z) resides in G, the matrix Lie group of g. This property is very relevant when solving Lie-group ODEs and is not usually fulfilled by standard approximations to the matrix exponential. We propose algorithms based on a splitting of Z into matrices having a very simple structure, usually one row and one column (or a few rows and a few columns), whose exponential is computed very cheaply to machine accuracy. The proposed methods have a complexity of O \Gamma n 3 \Delta , the constant is small, depending on the order and the Lie algebra g. The algorithms are recommended in cases where it is of fundamental importance that the approximation for the exponential resides in G, and when the order of approx...

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In this paper we explore the computation of the matrix exponential in a manner that is consistent with Lie-group structure. Our point of departure is the method of generalized polar decompositions, which we modify and combine with similarity transformations that bring the underlying matrix to a form more amenable to e#cient computation. We develop techniques valid for a range of Lie-groups: the orthogonal group, the symplectic group, Lorenz, isotropy and scaling groups. However, the GPD approach is equally promising in a more general context: even when Lie-group structure is not at issue, our algorithm is more efficient in many settings than classical methods for the computation of the matrix exponential.