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# Conditioning and relative error propagation in linear autonomous ordinary differential equations

• * Corresponding author

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.

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

 Citation: • • 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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