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September  2018, 23(7): 2879-2909. doi: 10.3934/dcdsb.2018165

Conditioning and relative error propagation in linear autonomous ordinary differential equations

Dipartimento di Matematica e Geoscienze, Università di Trieste, Via Valerio 12/1, 34127, Trieste, Italy

* Corresponding author

Received  April 2017 Revised  January 2018 Published  June 2018

Fund Project: The first author is supported in part by the GNCS of the italian "Istituto Nazionale di Alta Matematica".

In this paper, we study the relative error propagation in the solution of linear autonomous ordinary differential equations with respect to perturbations in the initial value. We also consider equations with a constant forcing term and a nonzero equilibrium. The study is carried out for equations defined by normal matrices.

Citation: Stefano Maset. Conditioning and relative error propagation in linear autonomous ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2879-2909. doi: 10.3934/dcdsb.2018165
References:
[1]

A. H. Al-Mohy and N. J. Higham, Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation, SIAM J. Matrix Anal. Appl., 30 (2009), 1639-1657.  doi: 10.1137/080716426.  Google Scholar

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F. Burgisser and F. Cucker, Condition, Springer 2013. doi: 10.1007/978-3-642-38896-5.  Google Scholar

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E. Hairer, S. Norsett and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, Springer-Verlag Berlin Heidelberg, Second Revised Edition, 1993.  Google Scholar

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N. J. Higham, Functions of Matrices, Theory and Computation, Siam, 2008. doi: 10.1137/1.9780898717778.  Google Scholar

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N. J. Higham and A. H. Al-Mohy, Computing matrix functions, Acta Numerica, 19 (2010), 159-208.  doi: 10.1017/S0962492910000036.  Google Scholar

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B. Kagstrom, Bounds and perturbations for the matrix exponential, BIT, 17 (1977), 39-57.   Google Scholar

show all references

References:
[1]

A. H. Al-Mohy and N. J. Higham, Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation, SIAM J. Matrix Anal. Appl., 30 (2009), 1639-1657.  doi: 10.1137/080716426.  Google Scholar

[2]

F. Burgisser and F. Cucker, Condition, Springer 2013. doi: 10.1007/978-3-642-38896-5.  Google Scholar

[3]

R. GroneC. R. JohnsonE. M. Sa and H. Wolkowicz, Normal matrices, Linear Algebra and its Applications, 87 (1987), 213-225.  doi: 10.1016/0024-3795(87)90168-6.  Google Scholar

[4]

G. Golub and C. F. Van Loan, Matrix Computations, The John Hopkins University Press, third edition 1996.  Google Scholar

[5]

E. Hairer, S. Norsett and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, Springer-Verlag Berlin Heidelberg, Second Revised Edition, 1993.  Google Scholar

[6]

N. J. Higham, Functions of Matrices, Theory and Computation, Siam, 2008. doi: 10.1137/1.9780898717778.  Google Scholar

[7]

N. J. Higham and A. H. Al-Mohy, Computing matrix functions, Acta Numerica, 19 (2010), 159-208.  doi: 10.1017/S0962492910000036.  Google Scholar

[8]

B. Kagstrom, Bounds and perturbations for the matrix exponential, BIT, 17 (1977), 39-57.   Google Scholar

Figure 1.  Propagation of absolute and relative errors for $y_0 = (0.9,-0.7)$ and $\widetilde{y}_0 = \left( 0.91,-0.71\right)$
Figure 2.  Propagation of absolute and relative errors for $y_0 = (0.9,-0.7)$ and $\widetilde{y}_0 = \left( 0.91,-0.69\right)$
Figure 3.  Propagation of absolute and relative errors for $y_0 = (1,-1)$ and $\widetilde{y}_0 = \left( 1.01,-0.99\right)$
Figure 4.  $2$-norm $\Vert y(t)\Vert_2$ and condition number $J_2(t,A,b,y_0)$ for $t\in[0,3T]$ in case of $\widehat{d}_{0} = \frac{1}{\sqrt{2}}\left( 0,1,1\right)$ and $c = 100$ (situation B)
Figure 5.  $2$-norm $\Vert y(t)\Vert_2$ and condition number $J_2(t,A,b,y_0)$ for $t\in[0,3T]$ in case of $\widehat{d}_{0} = \frac{1}{\sqrt{2.0202}}\left( 0.01,1.01,1\right)$ and $c = 1000$ (situation C)
Figure 6.  $2$-norm $\Vert y(t)\Vert_2$ and condition number $J_2(t,A,b,y_0)$ for $t\in[0,3T]$ in case of $\widehat{d}_{0}=\frac{1}{\sqrt{2}}\left( 0,1,1\right)$ and $c=100$ (situation B).
Figure 7.  $2$-norm $\Vert y(t)\Vert_2$ and condition number $J_2(t,A,b,y_0)$ for $t\in[0,3T]$ in case of $\widehat{d}_{0}=\frac{1}{\sqrt{2.0202}}\left( 0.01,1.01,1\right)$ and $c=1000$ (situation C).
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