The logarithmic norm. History and modern theory. (English) Zbl 1102.65088

Summary: In his 1958 thesis “Stability and error bounds”, Germund Dahlquist introduced the logarithmic norm in order to derive error bounds in initial value problems (IVPs), 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 boundary value problems, to the solvability of algebraic, nonlinear, operator, and functional equations.


65L70 Error bounds for numerical methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
65-03 History of numerical analysis
34A34 Nonlinear ordinary differential equations and systems
01A61 History of mathematics in the 21st century
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis


Full Text: DOI


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