Abstract
We define the Newton iteration for solving the equation f(y) = 0, where f is a map from a Lie group to its corresponding Lie algebra. Two versions are presented, which are formulated independently of any metric on the Lie group. Both formulations reduce to the standard method in the Euclidean case, and are related to existing algorithms on certain Riemannian manifolds. In particular, we show that, under classical assumptions on f, the proposed method converges quadratically. We illustrate the techniques by solving a fixed-point problem arising from the numerical integration of a Lie-type initial value problem via implicit Euler.
Similar content being viewed by others
REFERENCES
R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Springer-Verlag, New York, 1980.
V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., GTM no. 60, Springer-Verlag, New York, 1989.
R. L. Bryant, An introduction to Lie groups and symplectic geometry, in Geometry and Quantum Field Theory, D. S. Freed and K. K. Uhlenbeck, eds., AMS, IAS/Park City Math. Series vol. 1, 1995.
P. E. Crouch and R. Grossman, Numerical integration of ordinary differential equations on manifolds, J. Nonlinear Sci., 3 (1993) pp. 1–33.
A. Edelman, T. Arias and S. T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20 (1998), 303–353.
G. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
O. Gonzalez and A. M. Stuart, Remarks on the qualitative properties of modified equations, Tech. Report SCCM–96–03, Department of Mechanical Engineering, Stanford University, Stanford, CA.
S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978.
D. Higham, private communication.
A. Iserles, Solving linear ordinary differential equations by exponentials of iterated commutators, Numer. Math., 45 (1984), pp. 183–199.
S. Klarsfeld and J. A. Oteo, The Baker-Campbell-Hausdorff formula and the convergence of the Magnus expansion, J. Phys., A 22:21 (1989), pp. 4565–4572.
R. E. Mahony, The constrained Newton method on a Lie-group and the symmetric eigenvalue problem, Linear Algebra Appl., 248:1–3 (1996), pp. 67–89.
R. E. Mahony, Optimization Algorithms on Homogeneous Spaces: With Applications in Linear Systems Theory, PhD Thesis, Dept. of Systems Engineering, Australian National University, Canberra, 1994.
H. Munthe-Kaas, Runge-Kutta methods on Lie groups, Preprint, Department of Informatics, University of Bergen, 1996.
H. Munthe-Kaas, Lie-Butcher theory for Runge-Kutta methods, BIT, 35 (1996), pp. 572–587.
H. Munthe-Kaas and A. Zanna, Numerical integration of differential equations on homogeneous manifolds, Preprint, Department of Informatics, University of Bergen, 1996.
I. Najfeld and T. F. Havel, Derivatives of the matrix exponential and their computation, Adv. Appl. Math., 16:3 (1995), pp. 321–375.
B. Owren and A. Marthinsen, Runge-Kutta methods adapted to manifolds and based on rigid frames, BIT, 39 (1999), pp. 116–142.
B. Owren and B. Welfert, The Newton iteration on Lie groups, Technical Report 3/96, Department of Mathematical Sciences, NTNU, 1996.
V. S. Varadarajan, Lie Groups, Lie Algebras and their Representations, GTM no. 102, Springer-Verlag, New York, 1984.
F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, GTM no. 94, Springer-Verlag, New York, 1983.
Y.-P. Lin and H.L.-P. Hwang, Efficient computation of the matrix exponential using Padéapproximation, Computers & Chemistry, 16:4 (1992), pp. 285–293.
A. Zanna, The method of iterated commutators for ordinary differential equations on Lie groups, Tech. Report, University of Cambridge, DAMTP 1996/NA12, 1996.
A. Zanna, K. Engø, and H. Z. Munthe-Kaas, Adjoint and selfadjoint Lie-group methods, Tech. Report 1999/NA02, DAMTP, Cambridge University, 1999.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Owren, B., Welfert, B. The Newton Iteration on Lie Groups. BIT Numerical Mathematics 40, 121–145 (2000). https://doi.org/10.1023/A:1022322503301
Issue Date:
DOI: https://doi.org/10.1023/A:1022322503301