Conditioning and relative error propagation in linear autonomous ordinary differential equations

Stefano Maset

doi:10.3934/dcdsb.2018165

Article Contents

2018, Volume 23, Issue 7: 2879-2909. Doi: 10.3934/dcdsb.2018165

This issue Previous Article Underlying one-step methods and nonautonomous stability of general linear methods Next Article Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems

Conditioning and relative error propagation in linear autonomous ordinary differential equations

Stefano Maset^,

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

* Corresponding author
* Corresponding author

Received: March 31, 2017

Revised: December 31, 2017

Early access: May 2018

Published: July 2018

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

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Abstract

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.

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

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Figure 1. Propagation of absolute and relative errors for $y_0 = (0.9,-0.7)$ and $\widetilde{y}_0 = \left( 0.91,-0.71\right)$

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Figure 2. Propagation of absolute and relative errors for $y_0 = (0.9,-0.7)$ and $\widetilde{y}_0 = \left( 0.91,-0.69\right)$

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Figure 3. Propagation of absolute and relative errors for $y_0 = (1,-1)$ and $\widetilde{y}_0 = \left( 1.01,-0.99\right)$

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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)

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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)

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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).

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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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