A class of intrinsic schemes for orthogonal integration

SIAM Journal on Numerical Analysis

Volumn 40, Issue 6, 2002, Pages 2069-2084

  Celledoni, Elena ^a   Owren, Brynjulf ^b  

^a SINTEF   (Norway)

^b NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

Author keywords

Geometric integration; Homogeneous spaces; Numerical analysis; Numerical integration of ordinary differential equations on manifolds; Stiefel manifold; Time integration

Indexed keywords

APPROXIMATION THEORY; COMPUTATIONAL METHODS; COMPUTER PROGRAMMING; DIFFERENTIAL EQUATIONS; ENERGY CONSERVATION; ITERATIVE METHODS; LAGRANGE MULTIPLIERS; LYAPUNOV METHODS; MATRIX ALGEBRA; NUMERICAL METHODS; OPTIMIZATION;

GEOMETRIC INTEGRATION; HOMOGENEOUS SPACES; STIEFEL MANIFOLD; TIME INTEGRATION;

INTEGRATION;

EID: 1642544742     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142901385143     Document Type: Article

References (26)

1. U. ASCHER AND S. REICH, On some difficulties in integrating highly oscillatory Hamiltonian systems, in Computational Molecular Dynamics, Lecture Notes in Comput. Sci. Eng. 4, Springer-Verlag, Berlin, 1999, pp. 281-296.
2. T. J. BRIDGES AND S. REICH, Computing Lyapunov exponents on a Stiefel manifold, Phys. D, 156 (2001), pp. 219-238.
3. E. CELLEDONI AND B. OWREN, A Class of Low Complexity Intrinsic Schemes for Orthogonal Integration, Technical report Numerics 1/2001, The Norwegian University of Science and Technology, Trondheim, Norway, 2001.
4. E. CELLEDONI AND B. OWREN, On the Implementation of Lie Group Methods on the Stiefel Manifold, Technical report Numerics 9/2001, The Norwegian University of Science and Technology, Trondheim, Norway, 2001, Numer. Algorithms, submitted.
5. I. CHAVEL, Riemannian Geometry: A Modern Introduction, Cambridge Tracts in Math. 108, Cambridge University Press. Cambridge, UK, 1993.
6. P. E. CROUCH AND R. GROSSMAN, Numerical integration of ordinary differential equations on manifolds. J. Nonlinear Sci., 3 (1993). pp. 1-33.
7. L. DIECI, R. D. RUSSELL, AND S. VAN VLECK, On the computation of Lyapunov exponents for continuous dynamical systems, SIAM J. Numer. Anal., 34 (1997), pp. 402-423.
8. L. DIECI AND E. S. VAN VLECK, Computation of a few Lyapunov exponents for continuous and discrete dynamical systems, Appl. Numer. Math., 17 (1995), pp. 275-291.
9. L. DIECI AND E. S. VAN VLECK, Computation of orthonormal factors for fundamental solution matrices, Numer. Math., 83 (1999), pp. 599-620.
10. F. DIELE, L. LOPEZ, AND R. PELUSO, The Cayley transform in the numerical solution of unitary differential systems, Adv. Comput. Math., 8 (1998), pp. 317-334.
11. U. DRESSLER, Symmetry property of the Lyapunov spectra of a class of dissipative dynamical systems with viscous damping, Phys. Rev. A (3), 38 (1988), pp. 2103-2109.
12. A. EDELMAN, T. A. ARIAS, AND S. T. SMITH, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 303-353.
13. E. EICH-SOELLNER AND C. FÜHRER, Numerical Methods in Multibody Dynamics, European Consort. Math. Indust., B. G. Teubner, Stuttgart, 1998.
14. K. GEIST, U. PARLITZ, AND W. LAUTERBORN, Comparison of different methods for computing Lyapunov exponents, Progr. Theoret. Phys., 83 (1990), pp. 875-893.
15. G. H. GOLUB AND C. F. V. LOAN, Matrix Computations, The Johns Hopkins University Press, Baltimore, MD. 1996.
16. E. HAIRER, Symmetric projection methods for differential equations on manifolds, BIT, 40 (2000), pp. 726-734.
17. E. HAIRER, C. LUBICH, AND G. WANNER, Geometric Numerical Integration, Ser. Comput. Math. 31, Springer-Verlag, Berlin, 2002.
18. E. HAIRER AND G. WANNER, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, 2nd ed., Springer-Verlag, Berlin, 1996.
19. S. J. HAMMARLING, Numerical solution of the stable non-negative definite Lyapunov equation, IMA J. Numer. Anal., 2 (1982), pp. 303-323.
20. D. J. HIGHAM, Time-stepping and preserving orthonormality, BIT, 37 (1997), pp. 24-36.
21. H. MUNTHE-KAAS, High order Runge-Kutta methods on manifolds, Appl. Numer. Math., 29 (1999), pp. 115-127.
22. B. OWREN AND A. MARTHINSEN, Integration methods based on canonical coordinates of the second kind, Numer. Math., 87 (2001), pp. 763-790.
23. F. A. POTRA AND W. C. RHEINBOLDT, On the numerical solution of Euler-Lagrange equations, Mech. Structures Mach., 19 (1991), pp. 1-18.
24. F. A. POTRA AND J. YEN, Implicit numerical integration for Euler-Lagrange equations via tangent space parametrization, Mech. Structures Mach., 19 (1991), pp. 77-98.
25. M. SHUB, Some remarks on dynamical systems and numerical analysis, in Dynamical Systems and Partial Differential Equations, L. Lara-Carrero and J. Lewowicz, eds., Equinoccio, Caracas, Venezuela, 1986, pp. 69-92.
26. N. TRENDAFILOV, Projected Gradient Approach to Multivariate Data Analysis, A Review, http://people.mech.kuleuven.ac.be/̃ntrendaf//pubs.htm.