Methods for the approximation of the matrix exponential in a Lie‐algebraic setting

Discretization methods for ordinary differential equations based on the use of matrix exponentials have been known for decades. This set of ideas has come off age and acquired greater interest recently, within the context of geometric integration and discretization methods on manifolds based on the...

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Bibliographic Details
Published in:IMA journal of numerical analysis 2001-04, Vol.21 (2), p.463-488
Main Authors: Celledoni, Elena, Iserles, Arieh
Format: Article
Language:English
Subjects:
Exact sciences and technology
Group theory
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Numerical linear algebra
Ordinary differential equations
Sciences and techniques of general use
Topological groups, lie groups
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Summary:Discretization methods for ordinary differential equations based on the use of matrix exponentials have been known for decades. This set of ideas has come off age and acquired greater interest recently, within the context of geometric integration and discretization methods on manifolds based on the use of Lie‐group actions. In the present paper we study the approximation of the matrix exponential in a particular context: given a Lie group G and its Lie algebra g, we seek approximants F(t B) of exp(t B) such that F(t B) ∈ G if B ∈ g. Having fixed a basis V1, …, Vd of g, we write F(t B) as a composition of exponentials of the type exp(αi (t) Vi), where αi for i = 1, 2, …, d are scalar functions. In this manner it becomes possible to increase the order of the approximation without increasing the number of exponentials to evaluate and multiply together. We study order conditions and implementation details and conclude the paper with some numerical experiments.
ISSN:0272-4979
1464-3642
DOI:10.1093/imanum/21.2.463