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


[1] J. C. Butcher, A stability property of implicit Runge–Kutta methods, BIT, 15 (1975), pp. 358–361. · Zbl 0333.65031
[2] M. G. Crandall and T. Liggett, Generation of semigroups of nonlinear transformation on general Banach spaces, Am. J. Math., 93 (1971), pp. 265–298. · Zbl 0226.47038
[3] G. Dahlquist, Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations, Almqvist & Wiksells, Uppsala, 1958; Transactions of the Royal Institute of Technology, Stockholm, 1959. · Zbl 0085.33401
[4] G. Dahlquist, A special stability problem for linear multistep methods, BIT, 3 (1963), pp. 27–43. · Zbl 0123.11703
[5] G. Dahlquist, G-stability is equivalent to A-stability, BIT, 18 (1978), pp. 384–401. · Zbl 0413.65057
[6] G. Dahlquist, personal communication, c. 1985.
[7] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. · Zbl 0559.47040
[8] K. Dekker and J. G. Verwer, Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations, North Holland, New York, 1984. · Zbl 0571.65057
[9] C. Desoer and H. Haneda, The measure of a matrix as a tool to analyze computer algorithms for circuit analysis, IEEE Trans. Circuit Theory, 19 (1972), pp. 480–486.
[10] A. Dontchev and F. Lempio, Difference methods for differential inclusions: A survey, SIAM Rev., 34 (1992), pp. 263–294. · Zbl 0757.34018
[11] R. Frank, J. Schneid, and C. W. Ueberhuber, The concept of B-convergence, SIAM J. Numer. Anal., 18 (1981), pp. 753–780. · Zbl 0467.65032
[12] E. Hansen, Convergence of multistep time discretizations of nonlinear dissipative evolution equations, SIAM J. Numer. Anal., 44 (2006), pp. 55–65. · Zbl 1118.65055
[13] E. Hansen, Runge–Kutta time discretizations of nonlinear dissipative evolution equations, Math. Comp., 75 (2006), pp. 631–640. · Zbl 1098.65058
[14] I. Higueras and B. García-Celayeta, Logarithmic norms of matrix pencils, SIAM J. Matrix Anal. (1997), pp. 646–666.
[15] I. Higueras and G. Söderlind, Logarithmic norms and nonlinear DAE stability, BIT, 42 (2002), pp. 823–841. · Zbl 1019.65062
[16] J. F. B. M. Kraaijevanger, B-convergence of the implicit midpoint rule and the trapezoidal rule, BIT, 25 (1985), pp. 652–666. · Zbl 0584.65048
[17] S. M. Lozinskii, Error estimates for the numerical integration of ordinary differential equations, part I, Izv. Vyss. Uceb. Zaved Matematika, 6 (1958), pp. 52–90 (Russian).
[18] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. · Zbl 0241.65046
[19] L. F. Shampine, What is stiffness?, in Stiff Computation, R. C. Aiken, ed., Oxford, New York, 1985. · Zbl 0555.65050
[20] T. Ström, On logarithmic norms, SIAM J. Numer. Anal., 2 (1975), pp. 741–753. · Zbl 0321.15012
[21] G. Söderlind, Bounds on nonlinear operators in finite-dimensional Banach spaces, Numer. Math., 50 (1986), pp. 27–44. · Zbl 0585.47047
[22] M. Spijker, Contractivity in the numerical solution of initial value problems, Numer. Math., 42 (1983), pp. 271–290. · Zbl 0504.65030
[23] H.-J. Stetter, Analysis of Discretization Methods for Ordinary Differential Equations, Springer, Berlin, Heidelberg, New York, 1973. · Zbl 0276.65001
[24] L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press, Princeton, 2005. · Zbl 1085.15009
[25] J. von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr., 4 (1951), pp. 258–281. · Zbl 0042.12301
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