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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Underlying one-step methods and nonautonomous stability of general linear methods
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 |
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:
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
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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. |
| [2] |
F. Burgisser and F. Cucker, Condition, Springer 2013.
doi: 10.1007/978-3-642-38896-5. |
| [3] |
R. Grone, C. R. Johnson, E. 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.
|
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. |
| [2] |
F. Burgisser and F. Cucker, Condition, Springer 2013.
doi: 10.1007/978-3-642-38896-5. |
| [3] |
R. Grone, C. R. Johnson, E. 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.
|
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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