The logarithmic norm. History and modern theory
(2006) In BIT Numerical Mathematics 46(3). p.631-652- Abstract
- In his 1958 thesis Stability and Error Bounds, Germund Dahlquist introduced the logarithmic norm in order to derive error bounds in initial value problems, using differential inequalities that distinguished between forward and reverse time integration. Originally defined for matrices, the logarithmic norm can be extended to bounded linear operators, but the extensions to nonlinear maps and unbounded operators have required a functional analytic redefinition of the concept. This compact survey is intended as an elementary, but broad and largely self-contained, introduction to the versatile and powerful modern theory. Its wealth of applications range from the stability theory of IVPs and BVPs, to the solvability of algebraic, nonlinear,... (More)
- In his 1958 thesis Stability and Error Bounds, Germund Dahlquist introduced the logarithmic norm in order to derive error bounds in initial value problems, using differential inequalities that distinguished between forward and reverse time integration. Originally defined for matrices, the logarithmic norm can be extended to bounded linear operators, but the extensions to nonlinear maps and unbounded operators have required a functional analytic redefinition of the concept. This compact survey is intended as an elementary, but broad and largely self-contained, introduction to the versatile and powerful modern theory. Its wealth of applications range from the stability theory of IVPs and BVPs, to the solvability of algebraic, nonlinear, operator, and functional equations. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/393269
- author
- Söderlind, Gustaf LU
- organization
- publishing date
- 2006
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- difference method, differential inequality, monotonicity theorem, uniform, monotonicity, logarithmic norm, logarithmic Lipschitz constant, stability, error bound, Lax principle
- in
- BIT Numerical Mathematics
- volume
- 46
- issue
- 3
- pages
- 631 - 652
- publisher
- Springer
- external identifiers
-
- wos:000240721800014
- scopus:33749001694
- ISSN
- 0006-3835
- DOI
- 10.1007/s10543-006-0069-9
- language
- English
- LU publication?
- yes
- additional info
- The information about affiliations in this record was updated in December 2015. The record was previously connected to the following departments: Numerical Analysis (011015004)
- id
- 67c82d12-1002-46bd-8084-d81f7409792d (old id 393269)
- date added to LUP
- 2016-04-01 17:14:56
- date last changed
- 2022-03-23 00:11:26
@article{67c82d12-1002-46bd-8084-d81f7409792d,
abstract = {{In his 1958 thesis Stability and Error Bounds, Germund Dahlquist introduced the logarithmic norm in order to derive error bounds in initial value problems, using differential inequalities that distinguished between forward and reverse time integration. Originally defined for matrices, the logarithmic norm can be extended to bounded linear operators, but the extensions to nonlinear maps and unbounded operators have required a functional analytic redefinition of the concept. This compact survey is intended as an elementary, but broad and largely self-contained, introduction to the versatile and powerful modern theory. Its wealth of applications range from the stability theory of IVPs and BVPs, to the solvability of algebraic, nonlinear, operator, and functional equations.}},
author = {{Söderlind, Gustaf}},
issn = {{0006-3835}},
keywords = {{difference method; differential inequality; monotonicity theorem; uniform; monotonicity; logarithmic norm; logarithmic Lipschitz constant; stability; error bound; Lax principle}},
language = {{eng}},
number = {{3}},
pages = {{631--652}},
publisher = {{Springer}},
series = {{BIT Numerical Mathematics}},
title = {{The logarithmic norm. History and modern theory}},
url = {{http://dx.doi.org/10.1007/s10543-006-0069-9}},
doi = {{10.1007/s10543-006-0069-9}},
volume = {{46}},
year = {{2006}},
}