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The logarithmic norm. History and modern theory

Söderlind, Gustaf LU (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)
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author
organization
publishing date
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
id
67c82d12-1002-46bd-8084-d81f7409792d (old id 393269)
date added to LUP
2007-10-19 12:26:56
date last changed
2019-03-17 04:30:09
@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},
  keyword      = {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},
  volume       = {46},
  year         = {2006},
}