Recurrence relations and convergence theory of the generalized polar decomposition on Lie groups

Author:
Antonella Zanna

Journal:
Math. Comp. **73** (2004), 761-776

MSC (2000):
Primary 51A50; Secondary 65L99, 58A99

DOI:
https://doi.org/10.1090/S0025-5718-03-01602-8

Published electronically:
November 5, 2003

MathSciNet review:
2031405

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Abstract: The subject matter of this paper is the analysis of some issues related to *generalized polar decompositions* on Lie groups. This decomposition, depending on an involutive automorphism , is equivalent to a factorization of , being a Lie group, as with and , and was recently discussed by Munthe-Kaas, Quispel and Zanna together with its many applications to numerical analysis. It turns out that, contrary to , an analysis of is a very complicated task. In this paper we derive the series expansion for , obtaining an explicit recurrence relation that completely defines the function in terms of projections on a Lie triple system and a subalgebra of the Lie algebra , and obtain bounds on its region of analyticity. The results presented in this paper have direct application, among others, to linear algebra, integration of differential equations and approximation of the exponential.

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Additional Information

**Antonella Zanna**

Affiliation:
Institutt for informatikk, University of Bergen, Høyteknologisenteret, Thormøhlensgate 55, N-5020 Bergen, Norway

Email:
anto@ii.uib.no

DOI:
https://doi.org/10.1090/S0025-5718-03-01602-8

Keywords:
Lie group,
Lie algebra,
generalized polar decomposition,
generalized Cartan decomposition

Received by editor(s):
September 5, 2001

Received by editor(s) in revised form:
October 1, 2002

Published electronically:
November 5, 2003

Article copyright:
© Copyright 2003
American Mathematical Society