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Conditioning and relative error propagation in linear autonomous ordinary differential equations

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

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  • 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.

    Mathematics Subject Classification: Primary: 65F35, 65F60; Secondary: 34D10.

    Citation:

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  • 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).

  • [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.
    [2] F. Burgisser and F. Cucker, Condition, Springer 2013. doi: 10.1007/978-3-642-38896-5.
    [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.
    [4] G. Golub and C. F. Van Loan, Matrix Computations, The John Hopkins University Press, third edition 1996.
    [5] E. Hairer, S. Norsett and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, Springer-Verlag Berlin Heidelberg, Second Revised Edition, 1993.
    [6] N. J. Higham, Functions of Matrices, Theory and Computation, Siam, 2008. doi: 10.1137/1.9780898717778.
    [7] N. J. Higham and A. H. Al-Mohy, Computing matrix functions, Acta Numerica, 19 (2010), 159-208.  doi: 10.1017/S0962492910000036.
    [8] B. Kagstrom, Bounds and perturbations for the matrix exponential, BIT, 17 (1977), 39-57. 
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