Figure 1.
Propagation of absolute and relative errors for $y_0 = (0.9,-0.7)$ and $\widetilde{y}_0 = \left( 0.91,-0.71\right)$
-
Previous Article
Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems
- DCDS-B Home
- This Issue
-
Next Article
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
![]()
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).
| [1] |
Samuel Bernard, Fabien Crauste. Optimal linear stability condition for scalar differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1855-1876. doi: 10.3934/dcdsb.2015.20.1855 |
| [2] |
Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87 |
| [3] |
Zvi Artstein. Averaging of ordinary differential equations with slowly varying averages. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 353-365. doi: 10.3934/dcdsb.2010.14.353 |
| [4] |
Jehad O. Alzabut. A necessary and sufficient condition for the existence of periodic solutions of linear impulsive differential equations with distributed delay. Conference Publications, 2007, 2007 (Special) : 35-43. doi: 10.3934/proc.2007.2007.35 |
| [5] |
W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209 |
| [6] |
Aeeman Fatima, F. M. Mahomed, Chaudry Masood Khalique. Conditional symmetries of nonlinear third-order ordinary differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 655-666. doi: 10.3934/dcdss.2018040 |
| [7] |
Hongwei Lou, Weihan Wang. Optimal blowup/quenching time for controlled autonomous ordinary differential equations. Mathematical Control & Related Fields, 2015, 5 (3) : 517-527. doi: 10.3934/mcrf.2015.5.517 |
| [8] |
Jean Mawhin, James R. Ward Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 39-54. doi: 10.3934/dcds.2002.8.39 |
| [9] |
Iasson Karafyllis, Lars Grüne. Feedback stabilization methods for the numerical solution of ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 283-317. doi: 10.3934/dcdsb.2011.16.283 |
| [10] |
Alessandro Fonda, Fabio Zanolin. Bounded solutions of nonlinear second order ordinary differential equations. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 91-98. doi: 10.3934/dcds.1998.4.91 |
| [11] |
Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281 |
| [12] |
Ping Lin, Weihan Wang. Optimal control problems for some ordinary differential equations with behavior of blowup or quenching. Mathematical Control & Related Fields, 2018, 8 (3&4) : 809-828. doi: 10.3934/mcrf.2018036 |
| [13] |
Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods. Kinetic & Related Models, 2017, 10 (1) : 61-91. doi: 10.3934/krm.2017003 |
| [14] |
Lijuan Wang, Yashan Xu. Admissible controls and controllable sets for a linear time-varying ordinary differential equation. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1001-1019. doi: 10.3934/mcrf.2018043 |
| [15] |
Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905 |
| [16] |
Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299 |
| [17] |
Ben-Yu Guo, Zhong-Qing Wang. A spectral collocation method for solving initial value problems of first order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1029-1054. doi: 10.3934/dcdsb.2010.14.1029 |
| [18] |
Wen Li, Song Wang, Volker Rehbock. A 2nd-order one-point numerical integration scheme for fractional ordinary differential equations. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 273-287. doi: 10.3934/naco.2017018 |
| [19] |
Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229 |
| [20] |
Angelo Favini, Yakov Yakubov. Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 595-614. doi: 10.3934/dcdss.2011.4.595 |
2018 Impact Factor: 1.008
Tools
Metrics
Other articles
by authors
[Back to Top]