Computation of the exponential of large sparse skew-symmetric matrices

SIAM Journal on Scientific Computing

Volumn 27, Issue 1, 2005, Pages 278-293

  Del Buono, N ^a   Lopez, L ^a   Peluso, R ^b  

^a UNIVERSITY OF BARI   (Italy)

^b POLITECNICO DI BARI   (Italy)

Author keywords

Geometric integration of ordinary differential equations; Singular value decomposition; Skew symmetric matrices; Splitting techniques

Indexed keywords

INTEGRATION; ITERATIVE METHODS; MATRIX ALGEBRA; ORDINARY DIFFERENTIAL EQUATIONS;

GEOMETRIC INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS; SINGULAR VALUE DECOMPOSITION; SKEW-SYMMETRIC MATRICES; SPLITTING TECHNIQUES;

COMPUTATIONAL METHODS;

EID: 33144471106     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/030600758     Document Type: Article

References (28)

1. A. O. BARUT, J. R. ZENI, AND A. LAUFER, The exponential map for the conformal group O(2,4), J. Phys. A, 27 (1994), pp. 5239-5250.
2. M. BENZI AND G. H. GOLUB, Bounds for the entries of matrix functions with applications to preconditioning, BIT, 39 (1999), pp. 417-438.
3. L. BERGAMASCHI AND M. VIANELLO, Efficient computation of the exponential operator for large, sparse, symmetric matrices, Numer. Linear Algebra Appl., 7 (2000), pp. 27-45.
4. E. CELLEDONI, A note on the numerical integration of the KdV equation via isospectral deformations, J. Phys. A, 34 (2001), pp. 2205-2214.
5. E. CELLEDONI AND A. ISERLES, Approximating the exponential from a Lie algebra to a Lie group, Math. Comp., 69 (2000), pp. 1457-1480.
6. E. CELLEDONI AND A. ISERLES, Methods for the approximation of the matrix exponential in a Lie-algebraic setting, IMA J. Numer. Anal., 21 (2001), pp. 463-488.
7. J.-B. CHEN, H. MUNTHE-KAAS, AND M.-Z. QIN, Square-conservative schemes for a class of evolution equations using Lie-group methods, SIAM J. Numer. Anal., 39 (2002), pp. 2164-2178.
8. J. W. DANIEL, W. B. GRAGG, L. KAUFMAN, AND G. W. STEWART, Reorthogonalization and stable algorithms updating the Gram-Schmidt QR factorization, Math. Comp., 30 (1976), pp. 772-795.
9. J. DEMMEL AND W. KAHAN, Accurate singular values of bidiagonal matrices, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 873-912.
10. K. V. FERNANDO AND B. N. PARLETT, Accurate singular values and differential qd algorithms, Numer. Math., 67 (1994), pp. 191-229.
11. E. GALLOPOULOS AND Y. SAAD, Efficient solution of parabolic equations by Krylov approximation methods, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 1236-1264.
12. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computation, 3rd ed., The John Hopkins University Press, Baltimore, 1996.
13. E. HAIRER, C. LUBICH, AND G. WANNER, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer-Verlag, Berlin, 2002.
14. M. HOCHBRUCK AND C. LUBICH, On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 34 (1997) pp. 1911-1925.
15. R. HORN AND C. JOHNSON, Topics in Matrix Analysis, Cambridge University Press, New York, 1995.
16. W. H. HUNDSDORFER, Numerical Solution of Advection-Diffusion-Reaction Equations, CWI Report NM-N9603, Center for Mathematics and Computer Science CWI, Amsterdam, 1996.
17. A. ISERLES, H. MUNTHE-KAAS, S. NØRSETT, AND A. ZANNA, Lie-group methods, in Acta Numerica, Acta Numer. 9, Cambridge University Press, Cambridge, UK, 2000, pp. 215-365.
18. A. ISERLES, How large is the exponential of a banded matrix?, New Zealand J. Math., 29 (2000), pp. 177-192.
19. A. ISERLES AND A. ZANNA, Efficient computation of the matrix exponential by generalized polar decompositions, SIAM J. Numer. Anal., 42 (2005), pp. 2218-2256.
20. A. ISERLES, E. MARTHINSEN, AND S. NØRSETT, On the implementation of the method of Magnus series for linear differential equations, BIT, 39 (1999), pp. 281-304.
21. F. S. LEITE AND P. CROUCH, Closed forms for the exponential mapping on matrix Lie groups based on Putzer's method, J. Math. Phys., 40 (1999), pp. 3561-3568.
22. B. LEIMKUHLER, Adaptive geometric integrators based on splitting, scaling, and switching, in International Conference on Differential Equations, Vol. 1, 2, World Scientific, River Edge, NJ, 2000, pp. 988-993.
23. B. N. PARLETT, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980.
24. T. POLITI, A formula for the exponential of a real skew symmetric matrix of order 4, BIT, 41 (2001), pp. 842-845.
25. Y. SAAD, Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 29 (1992), pp. 209-228.
26. N. J. ZABUSKY AND M. D. KRUSKAL, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), pp. 240-243.
27. A. ZANNA, The Method of Iterated Commutators for Ordinary Differential Equations on Lie Groups, DAMTP Technical report NA1996/12, University of Cambridge, Cambridge, UK, 1996.
28. A. ZANNA AND H. Z. MUNTHE-KAAS, Generalized polar decompositions for the approximation of the matrix exponential, SIAM J. Matrix Anal. Appl., 23 (2002), pp. 840-862.