Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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
Published electronically: March 15, 2000
MathSciNet review: 1709149
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

Consider a differential equation $y^{'}=A(t,y)y, y(0)=y_0$ with $y_ 0\in \mathrm{G}$and $A:\mathbb{R}^{+}\times \mathrm{G}\rightarrow \mathfrak{g}$, where $\mathfrak{g}$ is a Lie algebra of the matricial Lie group $\mathrm{G}$. Every $B\in \mathfrak{g}$ can be mapped to $\mathrm{G}$ by the matrix exponential map $\operatorname{exp}{(tB)}$ with $t\in \mathbb{R}$.

Most numerical methods for solving ordinary differential equations (ODEs) on Lie groups are based on the idea of representing the approximation $y_n$ of the exact solution $y (t_n)$, $t_n \in \mathbb{R}^{+}$, by means of exact exponentials of suitable elements of the Lie algebra, applied to the initial value $y_0$. This ensures that $y_n\in \mathrm{G}$.

When the exponential is difficult to compute exactly, as is the case when the dimension is large, an approximation of $\operatorname{exp}{(tB)}$ 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 $\operatorname{exp}{(tB)}$ 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 $\mathfrak{g}$ and $\mathrm{G}$ are considered, in tandem with general theory. Order conditions are discussed, and a number of numerical experiments conclude the paper.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (1991): 65D15, 22E99, 65F30

Retrieve articles in all journals with MSC (1991): 65D15, 22E99, 65F30


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: http://dx.doi.org/10.1090/S0025-5718-00-01223-0
PII: S 0025-5718(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