In this paper, we study the relative error in the numerical solution of a linear ordinary differential equation y′(t) = Ay(t), t ≥ 0, where A is a normal matrix. The numerical solution is obtained by using at any step an approximation of the matrix exponential, e.g. a polynomial or a rational approximation. The error of the numerical solution with respect to the exact solution is due to this approximation as well as to a possible perturbation in the initial value. For an unperturbed initial value, we find: 1) unlike the absolute error, the relative error always grows linearly in time; 2) in the long-time, the contributions to the relative error relevant to non-rightmost eigenvalues of A disappear.

Relative error analysis of matrix exponential approximations for numerical integration

Stefano Maset
2021-01-01

Abstract

In this paper, we study the relative error in the numerical solution of a linear ordinary differential equation y′(t) = Ay(t), t ≥ 0, where A is a normal matrix. The numerical solution is obtained by using at any step an approximation of the matrix exponential, e.g. a polynomial or a rational approximation. The error of the numerical solution with respect to the exact solution is due to this approximation as well as to a possible perturbation in the initial value. For an unperturbed initial value, we find: 1) unlike the absolute error, the relative error always grows linearly in time; 2) in the long-time, the contributions to the relative error relevant to non-rightmost eigenvalues of A disappear.
2021
22-nov-2020
Pubblicato
https://www.degruyter.com/document/doi/10.1515/jnma-2020-0019/html
File in questo prodotto:
File Dimensione Formato  
Jnl of Numerical Mathematics 2021.pdf

Open Access dal 01/07/2022

Descrizione: Articolo
Tipologia: Documento in Versione Editoriale
Licenza: Copyright Editore
Dimensione 5.29 MB
Formato Adobe PDF
5.29 MB Adobe PDF Visualizza/Apri
Jnl+of+Numerical+Mathematics+2021-Post_print.pdf

Open Access dal 04/07/2022

Tipologia: Bozza finale post-referaggio (post-print)
Licenza: Digital Rights Management non definito
Dimensione 5.86 MB
Formato Adobe PDF
5.86 MB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11368/2978075
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 4
  • ???jsp.display-item.citation.isi??? 3
social impact