Abstract
In this paper, we study the relative error in the numerical solution of a linear ordinary differential equation y'(t) = Ay(t), t ≥ 0, where A is a normal matrix. The numerical solution is obtained by using at any step an approximation of the matrix exponential, e.g., a polynomial or a rational approximation. The error of the numerical solution with respect to the exact solution is due to this approximation as well as to a possible perturbation in the initial value. For an unperturbed initial value, we have found: (1) unlike the absolute error, the relative error always grows linearly in time; (2) in the long-time, the contributions to the relative error relevant to non-rightmost eigenvalues of A disappear.
Funding: The author thanks INdAM-GNCS and the University of Trieste for the financial support.
References
[1] F. Bürgisser and F. Cucker, Condition. The Geometry of Numerical Algorithms, Springer, 2013.10.1007/978-3-642-38896-5Search in Google Scholar
[2] L. Dieci and A. Papini, Pade approximation for the exponential of a block triangular matrix, Linear Algebra Appl., 308 (2000), 183-202.10.1016/S0024-3795(00)00042-2Search in Google Scholar
[3] L. Dieci and L. Lopez, Numerical integration of networks of differential equations. Submitted.Search in Google Scholar
[4] B. Kagstrom, Bounds and perturbations for the matrix exponential, BIT, 17 (1977), 39-57.10.1007/BF01932398Search in Google Scholar
[5] N. J. Higham, Functions of Matrices. Theory and Computation, SIAM, 2008.10.1137/1.9780898717778Search in Google Scholar
[6] N. J. Higham, The scaling and squaring method for the matrix exponential revisited, SIAM Review, 51 (2009), 747-764.10.1137/090768539Search in Google Scholar
[7] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff Differential-Algebraic Problems. Springer-Verlag, Berlin-Heidelberg, 1996.10.1007/978-3-642-05221-7Search in Google Scholar
[8] A. Iserles and S. Norsett. Order Stars: Theory and Applications, Chapman and Hall, 1991.10.1007/978-1-4899-3071-2Search in Google Scholar
[9] S. Maset, Conditioning and relative error propagation in linear autonomous ordinary differential equations, Discrete and Continuous Dynamical Systems B, 23 (2018), 2879-2909.10.3934/dcdsb.2018165Search in Google Scholar
[10] S. Maset, Relative error long-time behavior in matrix exponential approximations for numerical integration: the stiff situation. Submitted.Search in Google Scholar
[11] S. Maset, Further results on the relative error analysis of matrix exponential approximations for numerical integration. In preparation.Search in Google Scholar
[12] S. Maset, Relative error stability and instability of matrix exponential approximations for stiff numerical integration of long-time solutions, J. Comp. Appl. Math., bf390 (2021), 113387.10.1016/j.cam.2021.113387Search in Google Scholar
[13] C. Moler and C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Review, 45 (2003), 3-49.10.1137/S00361445024180Search in Google Scholar
[14] F. Morbidi, The deformed consensus protocol, Automatica, 49 (2013), 3049-3055.10.1016/j.automatica.2013.07.006Search in Google Scholar
[15] L. Shampine, I. Gladwell, and S. Thompson, Solving ODEs with MATLAB, Cambridge University Press, 2003.10.1017/CBO9780511615542Search in Google Scholar
[16] D. Thanou, X. Dong, D. Kressner, and P. Frossard, Learning heat diffusion graphs, IEEE Trans. Signal Inform. Processing Networks, 3 (2017), 484-499.10.1109/TSIPN.2017.2731164Search in Google Scholar
[17] R. Olfati-Saber and R. M. Murray, Consensus protocols for networks of dynamic agents, Proc. American Control Conference, 2 (2003), 951-956.Search in Google Scholar
[18] R. Olfati-Saber and J. S. Shamma, Consensus filters for sensor networks and distributed sensor fusion. In: Proc. of the 44th IEEE Conf. on Decision and Control, and the European Control Conference, CDC-ECC’05, 2005, pp. 6698-6703.10.1109/CDC.2005.1583238Search in Google Scholar
[19] R. Olfati-Saber, J. A. Fax, and R. M. Murray, Consensus and cooperation in networked multi-agent systems. In: Proc. of the IEEE, 95 (2007), 215-233.10.1109/JPROC.2006.887293Search in Google Scholar
[20] G. Wanner, E. Hairer, and S. Norsett, Order stars and stability theorems, BIT, 18 (1978), 475-489.10.1007/BF01932026Search in Google Scholar
[21] R. Ward, Numerical computation of the matrix exponential with accuracy estimates, SIAM J. Numer. Anal., 14 (1977), 600-610.10.1137/0714039Search in Google Scholar
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