Approximating the exponential from a Lie algebra to a Lie group

Authors:
Elena Celledoni and Arieh Iserles

Journal:
Math. Comp. **69** (2000), 1457-1480

MSC (1991):
Primary 65D15; Secondary 22E99, 65F30

DOI:
https://doi.org/10.1090/S0025-5718-00-01223-0

Published electronically:
March 15, 2000

MathSciNet review:
1709149

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Abstract | References | Similar Articles | Additional Information

Consider a differential equation with and , where is a Lie algebra of the matricial Lie group . Every can be mapped to by the matrix exponential map with .

Most numerical methods for solving ordinary differential equations (ODEs) on Lie groups are based on the idea of representing the approximation of the exact solution , , by means of exact exponentials of suitable elements of the Lie algebra, applied to the initial value . This ensures that .

When the exponential is difficult to compute exactly, as is the case when the dimension is large, an approximation of plays an important role in the numerical solution of ODEs on Lie groups. In some cases rational or polynomial approximants are unsuitable and we consider alternative techniques, whereby is approximated by a product of simpler exponentials.

In this paper we present some ideas based on the use of the Strang splitting for the approximation of matrix exponentials. Several cases of and are considered, in tandem with general theory. Order conditions are discussed, and a number of numerical experiments conclude the paper.

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Additional Information

**Elena Celledoni**

Affiliation:
DAMTP, Cambridge University, Silver Street, England CB3 9EW

Address at time of publication:
Department of Mathematical Sciences, NTNU 7491 Trondheim, Norway

Email:
elenac@math.ntnu.no

**Arieh Iserles**

Affiliation:
DAMTP, Cambridge University, Silver Street, England CB3 9EW

Email:
A.Iserles@damtp.cam.ac.uk

DOI:
https://doi.org/10.1090/S0025-5718-00-01223-0

Received by editor(s):
January 1, 1998

Received by editor(s) in revised form:
October 27, 1998

Published electronically:
March 15, 2000

Article copyright:
© Copyright 2000
American Mathematical Society