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The logarithmic norm. History and modern theory

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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.

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Authors and Affiliations

  1. Numerical Analysis, Centre for Mathematical Sciences, Lund University, Box 118, S-221 00, Lund, Sweden

    Gustaf Söderlind

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  1. Gustaf Söderlind
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Correspondence to Gustaf Söderlind.

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In memory of Germund Dahlquist (1925–2005).

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65L05

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Söderlind, G. The logarithmic norm. History and modern theory . Bit Numer Math 46, 631–652 (2006). https://doi.org/10.1007/s10543-006-0069-9

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  • DOI: https://doi.org/10.1007/s10543-006-0069-9

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