Abstract
This paper concerns with the numerical solution of a class of ordinary differential equations on G l(n), the set of all n×n nonsingular real matrices. In particular, we consider matrix dynamical systems of the form Y′=Y −T F(Y). The presence of the inverse of the solution and of possible escape times make the numerical solution of this kind of differential equations somewhat worrisome. Here, we suggest some numerical techniques to avoid some problems arising in its numerical solution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M.T. Chu, Inverse eigenvalue problems, SIAM Review 40 (1998) 1–39.
N. Del Buono and L. Lopez, Geometric integration on manifold of square oblique rotation matrices, SIAM J. Matrix Anal. Appl. 23(4) (2002) 974–989.
L. Dieci and T. Eirola, On smooth decompositions of matrices, SIAM J. Matrix Anal. Appl. 20(3) (1999) 800–819.
L. Dieci, D. Russell and E. Van Vleck, Unitary integration and applications to continuous orthonormalization techniques, SIAM J. Numer. Anal. 31 (1994) 261–281.
F. Diele, L. Lopez and R. Peluso, The Cayley transform in the numerical solution of unitary differential systems, Adv. Comput. Math. 8 (1998) 317–334.
E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, 2nd revised ed. (Springer, Berlin, 1993).
U. Helmke and J.B. Moore, Optimization and Dynamical Systems (Springer, London, 1994).
A. Iserles, H.Z. Munthe-Kaas, S.P. Nørsett and A. Zanna, Lie-group methods, CUP, Acta Numerica 9 (2000) 215–365.
N.T. Trendafilov and R.A. Lippert, The multi-mode Procrustes problem, Linear Algebra Appl. 349(1–3) (2002) 245–264.
K. Write, Differential equations for the analytic singular value decomposition of a matrix, Numer. Math. 63 (1992) 283.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Del Buono, N., Lopez, L. Numerical Integration of a Class of Ordinary Differential Equations on the General Linear Group of Matrices. Numerical Algorithms 34, 271–281 (2003). https://doi.org/10.1023/B:NUMA.0000005404.80948.ae
Issue Date:
DOI: https://doi.org/10.1023/B:NUMA.0000005404.80948.ae