Approximating the exponential from a Lie algebra to a Lie group
Authors:
Elena Celledoni and Arieh Iserles
Journal:
Math. Comp. 69 (2000), 14571480
MSC (1991):
Primary 65D15; Secondary 22E99, 65F30
Published electronically:
March 15, 2000
MathSciNet review:
1709149
Fulltext PDF Free Access
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Abstract: 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:
http://dx.doi.org/10.1090/S0025571800012230
PII:
S 00255718(00)012230
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
