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Interpolation in Special Orthogonal Groups
, 2007
"... The problem of constructing smooth interpolating curves in non-Euclidean spaces finds applications in different areas of science. In this paper we propose a scheme to generate interpolating curves in Lie groups, focusing on a special orthogonal group SO(n). Our technique is based on the exponential ..."
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The problem of constructing smooth interpolating curves in non-Euclidean spaces finds applications in different areas of science. In this paper we propose a scheme to generate interpolating curves in Lie groups, focusing on a special orthogonal group SO(n). Our technique is based on the exponential representation of elements of the group, which allows to transfer the problem to the corresponding Lie algebra so(n) and benefit from the linearity of this space. Due to the exponential representation we can obtain a high degree of smoothness of an interpolating curve at relatively low costs. The underlying problem is challenging because the standard SO(n) −→ so(n) map is multi-valued. 1
Explicit Magnus expansions for nonlinear equations
"... Abstract. In this paper we develop and analyse new explicit Magnus expansions for the nonlinear equation Y ′ = A(t, Y)Y defined in a matrix Lie group. In particular, integration methods up to order four are presented in terms of integrals which can be either evaluated exactly or replaced by conveni ..."
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Abstract. In this paper we develop and analyse new explicit Magnus expansions for the nonlinear equation Y ′ = A(t, Y)Y defined in a matrix Lie group. In particular, integration methods up to order four are presented in terms of integrals which can be either evaluated exactly or replaced by conveniently adapted quadrature rules. The structure of the algorithm allows to change the step size and even the order during the integration process, thus improving its efficiency. Several examples are considered, including isospectral flows and highly-oscillatory nonlinear differential equations. AMS classification scheme numbers: 65L05, 41A55, 22E60 § To whom correspondence should be addressed Explicit Magnus expansions for nonlinear equations 2 1.
