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147
Means and Averaging in the Group of Rotations
, 2002
"... 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 ..."
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Cited by 114 (3 self)
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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 closedform solutions of both notions of mean.
A Schur–Parlett algorithm for computing matrix functions
 SIAM J. Matrix Anal. Appl
"... Abstract. An algorithm for computing matrix functions is presented. It employs a Schur decomposition with reordering and blocking followed by the block form of a recurrence of Parlett, with functions of the nontrivial diagonal blocks evaluated via a Taylor series. A parameter is used to balance the ..."
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Cited by 75 (23 self)
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Abstract. An algorithm for computing matrix functions is presented. It employs a Schur decomposition with reordering and blocking followed by the block form of a recurrence of Parlett, with functions of the nontrivial diagonal blocks evaluated via a Taylor series. A parameter is used to balance the conflicting requirements of producing small diagonal blocks and keeping the separations of the blocks large. The algorithm is intended primarily for functions having a Taylor series with an infinite radius of convergence, but it can be adapted for certain other functions, such as the logarithm. Novel features introduced here include a convergence test that avoids premature termination of the Taylor series evaluation and an algorithm for reordering and blocking the Schur form. Numerical experiments show that the algorithm is competitive with existing specialpurpose algorithms for the matrix exponential, logarithm, and cosine. Nevertheless, the algorithm can be numerically unstable with the default choice of its blocking parameter (or in certain cases for all choices), and we explain why determining the optimal parameter appears to be a very difficult problem. A MATLAB implementation is available that is much more reliable than the function funm in MATLAB 6.5 (R13).
A Lie group variational integrator for the attitude dynamics of a rigid body with applications to the 3D pendulum
 Proceedings of the IEEE Conference on Control Applications
, 2005
"... 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 propert ..."
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Cited by 49 (30 self)
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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.
Lie group variational integrators for the full body problem
, 2007
"... We develop the equations of motion for full body models that describe the dynamics of rigid bodies, acting under their mutual gravity. The equations are derived using a variational approach where variations are defined on the Lie group of rigid body configurations. Both continuous equations of motio ..."
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Cited by 45 (28 self)
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We develop the equations of motion for full body models that describe the dynamics of rigid bodies, acting under their mutual gravity. The equations are derived using a variational approach where variations are defined on the Lie group of rigid body configurations. Both continuous equations of motion and variational integrators are developed in Lagrangian and Hamiltonian forms, and the reduction from the inertial frame to a relative frame is also carried out. The Lie group variational integrators are shown to be symplectic, to preserve conserved quantities, and to guarantee exact evolution on the configuration space. One of these variational integrators is used to simulate the dynamics of two rigid dumbbell bodies.
Nonsmooth Lagrangian mechanics and variational collision integrators
 SIAM Journal on Applied Dynamical Systems
, 2003
"... Abstract. Variational techniques are used to analyze the problem of rigidbody 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 Noetherstyle momentum ..."
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Cited by 40 (9 self)
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Abstract. Variational techniques are used to analyze the problem of rigidbody 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 Noetherstyle 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 symplecticmomentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the longtime stable energy behavior typical of variational methods is demonstrated.
Newton’s method on Riemannian manifolds: convariant alpha theory
 IMA J. Numer. Anal
, 2003
"... 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 mani ..."
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Cited by 40 (2 self)
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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
Hamilton–Pontryagin integrators on Lie groups
, 2007
"... Abstract In this paper, structurepreserving timeintegrators for rigid bodytype mechanical ..."
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Cited by 37 (7 self)
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Abstract In this paper, structurepreserving timeintegrators for rigid bodytype mechanical
On Magnus Integrators for TimeDependent Schrödinger Equations
 SIAM J. Numer. Anal
, 2002
"... Numerical methods based on the Magnus expansion are an ecient class of integrators for Schrodinger equations with timedependent 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 f ..."
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Cited by 34 (2 self)
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Numerical methods based on the Magnus expansion are an ecient class of integrators for Schrodinger equations with timedependent 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 optimalorder 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.
On the Global Error of Discretization Methods for HighlyOscillatory Ordinary Differential Equations
, 2000
"... Commencing from a globalerror formula, originally due to Henrici, we investigate the accumulation of global error in the numerical solution of linear highlyoscillating 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 globalerror ..."
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Cited by 30 (6 self)
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Commencing from a globalerror formula, originally due to Henrici, we investigate the accumulation of global error in the numerical solution of linear highlyoscillating 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 globalerror envelope for RungeKutta and Magnus methods. Our results are closely matched by numerical experiments. Motivated by the superior performance of Liegroup methods, we present a modification of the Magnus expansion which displays even better longterm behaviour in the presence of oscillations.