Remote Access Mathematics of Computation
Green Open Access

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


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?)

  • 1. R.V. Chacon and A.T. Fomenko.
    Recursion formulas for the Lie integral.
    Advances in Math., 88:200-257, 1991. MR 93e:22013
  • 2. P. E. Crouch and R. Grossman.
    Numerical Integration of Ordinary Differential Equations on Manifolds.
    J. Nonlinear Sci., 3:1-33, 1993. MR 94e:65069
  • 3. K. Feng and Z.-J. Shang.
    Volume-preserving algorithms for source-free dynamical systems.
    Numer. Math., 71:451-463, 1995. MR 96g:65065
  • 4. E. Forest.
    Sixth-order Lie group integrators.
    J. of Comp. Phys., 99:209-213, 1992. MR 93b:70003
  • 5. E. Gallopoulos and Y. Saad.
    Efficient solution of parabolic equations by Krylov approximation methods.
    SIAM J. Sci. Statist. Comput., 13:1236-1264, 1992. MR 93d:65085
  • 6. M. Hochbruck and Ch. Lubich.
    On Krylov Subspace Approximations to the Matrix Exponential Operator.
    SIAM J. Numer. Anal., 34:1911-1925, 1997. MR 98h:65018
  • 7. A. Iserles and S. P. Nørsett.
    On the Solution of Linear Differential Equations in Lie Groups. Roy. Soc. London Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357 (1999), 983-1019. CMP 99:13
  • 8. R. I. McLachlan.
    On the numerical integration of ordinary differential equations by symmetric composition methods.
    SIAM J. Sci. Comput., 16:151-168, 1995. MR 95j:65073
  • 9. H. Munthe-Kaas.
    Runge-Kutta Methods on Lie Groups.
    BIT, 38:92-111, 1998. MR 99f:65117
  • 10. H. Munthe-Kaas and A. Zanna.
    Numerical integration of differential equations on homogeneous manifolds.
    In F. Cucker and M. Shub, editors, Foundations of Computational Mathematics, pages 305-315. Springer Verlag, 1997. MR 99i:65082
  • 11. J. M. Sanz-Serna and M. P. Calvo.
    Numerical Hamiltonian Problems.
    AMMC 7. Chapman & Hall, London, 1994. MR 95f:65006
  • 12. M. Schatzman.
    Higher order alternate directions methods.
    Technical report, Université de Lyon, Laboratoire d'analyse numérique, 1997.
  • 13. Q. Sheng.
    Solving partial differential equations by exponential splitting.
    PhD thesis, Cambridge University, 1989.
  • 14. G. Strang.
    Accurate partial difference methods. II. Nonlinear problems.
    Num. Math., 6:37-49, 1964. MR 29:4215
  • 15. M. Suzuki.
    General theory of fractal path integrals with applications to many-body theories and statistical physics.
    J. of Math. Phys., 32:400-407, 1991. MR 92k:81096
  • 16. V. S. Varadarajan.
    Lie Groups, Lie Algebras, and Their Representation.
    GTM 102. Springer-Verlag, 1984. MR 85e:22001
  • 17. H. Yoshida.
    Construction of higher order symplectic integrators.
    Physics Letters A, 150:262-268, 1990. MR 91h:70014
  • 18. A. Zanna.
    The method of iterated commutators for ordinary differential equations on Lie groups.
    Technical Report 1996/NA12, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England, 1996.

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

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

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

American Mathematical Society