Nothing Special   »   [go: up one dir, main page]

skip to main content
research-article

On Logarithmic Norms

Published: 01 October 1975 Publication History

Abstract

This paper discusses some normlike properties of the logarithmic norm and related efficiency questions. Some computational questions are raised and some examples of use are indicated.

References

[1]
F. L. Bauer, J. Stoer, C. Witzgall, Absolute and monotonic norms, Numer. Math., 3 (1961), 257–264
[2]
R. Bellman, Stability Theory of Differential Equations, McGraw-Hill, New York, 1965
[3]
Richard Bellman, Introduction to matrix analysis, McGraw-Hill Book Co., Inc., New York, 1960xx+328
[4]
G. J. Cooper, Error bounds for numerical solutions of ordinary differential equations, Numer. Math., 18 (1971/72), 162–170
[5]
W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Co., Boston, Mass., 1965viii+166
[6]
G. Dahlquist, Stability and error bounds in the numerical integration of ordinary differential equations, Diss., (1958), reprinted in Trans. Royal Inst. of Technology, No. 130, Stockholm, Sweden, 1959
[7]
Germund G. Dahlquist, D. Greenspan, On rigorous error bounds in the numerical solution of ordinary differential equationsNumerical Solutions of Nonlinear Differential Equations (Proc. Adv. Sympos., Madison, Wis., 1966), John Wiley & Sons Inc., New York, 1966, 89–96
[8]
G. Dahlquist, 1966, Unpublished lecture notes, Atlas Computer Labs., Didcot, England
[9]
Charles A. Desoer, Hiromasa Haneda, The measure of a matrix as a tool to analyze computer algorithms for circuit analysis, IEEE Trans. Circuit Theory, CT-19 (1972), 480–486
[10]
Bo Einarsson, Numerical treatment of integro-differential equations with a certain maximum property, Numer. Math., 18 (1971/72), 267–288
[11]
L. Elsner, Verfahren zur Berechnung des Spektralradius nichtnegativer irreduzibler Matrizen, Computing (Arch. Elektron. Rechnen), 8 (1971), 32–39
[12]
John H. George, Robert W. Gunderson, Conditioning of linear boundary value problems, Nordisk Tidskr. Informationsbehandling (BIT), 12 (1972), 172–181
[13]
K. G. Guderley, C. L. Keller, A basic theorem in the computation of ellipsoidal error bounds, Numer. Math., 19 (1972), 218–229
[14]
Peter Henrici, Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices, Numer. Math., 4 (1962), 24–40
[15]
Peter Henrici, Problems of stability and error propagation in the numerical integration of ordinary differential equationsProc. Internat. Congr. Mathematicians (Stockholm 1962), Inst. Mittag-Leffler, Djursholm, 1963, 102–113
[16]
Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964xi+257
[17]
W. Kahan, An ellipsoidal error bound for linear systems of ordinary differential equations, SIAM Rev., 8 (1966), 568–569
[18]
S. M. Lozinskij, Error estimate for numerical integration of ordinary differential equations, Part 1, Izv. Vyss Ucebn. Zaved. Matematika, 6 (1958), 52–90, and Errata 5, 12 (1959), p. 222, no. 5. (In Russian.)
[19]
S. M. Lozinskij, A contribution to the theory of f nice matrices, Soviet Math. Dokl., 163 (1965), 1014–1016
[20]
S. M. Lozinskij, Estimates of a spherical matrix norm and of the corresponding logarithmic norm, Soviet Math. Dokl., 165 (1965), 1511–1514
[21]
E. E. Osborne, On pre-conditioning of matrices, J. Assoc. Comput. Mach., 7 (1960), 338–345
[22]
C. V. Pao, Logarithmic derivates of a square matrix, Linear Algebra and Appl., 6 (1973), 159–164
[23]
C. V. Pao, A further remark on the logarithmic derivatives of a square matrix, Linear Algebra and Appl., 7 (1973), 275–278
[24]
B. Parlett, C. Reinsch, Balancing a matrix for calculation of eigenvalues and eigenvectors, Numer. Math., 13 (1969), 293–304, Also in Handbook for Automatic Computation, vol. 2, Linear Algebra, Springer-Verlag, Berlin, 1971.
[25]
Torsten Ström, On logarithmic norms, Department of Information Processing and Computer Science, The Royal Institute of Technology, Stockholm, 1969i+8
[26]
T. Ström, On the use of majorants for strict error estimation of numerical solutions to ODE's, Rep., NA 70.10, Dept. Computer Science, Royal Inst. Tech., Stockholm, Sweden, 1970
[27]
T. Ström, Minimization of norms and logarithmic norms by diagonal similarities, Computing (Arch. Elektron. Rechnen), 10 (1972), 1–7
[28]
J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965xviii+662, London

Cited By

View all
  • (2024)Unveiling Delay Effects in Traffic Forecasting: A Perspective from Spatial-Temporal Delay Differential EquationsProceedings of the ACM Web Conference 202410.1145/3589334.3645688(1035-1044)Online publication date: 13-May-2024
  • (2024)On the maximum principle and high-order, delay-free integrators for the viscous Cahn–Hilliard equationAdvances in Computational Mathematics10.1007/s10444-024-10143-650:3Online publication date: 3-May-2024

Recommendations

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis  Volume 12, Issue 5
Oct 1975
145 pages
ISSN:0036-1429
DOI:10.1137/sjnaam.1975.12.issue-5
Issue’s Table of Contents

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 October 1975

Qualifiers

  • Research-article

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • Downloads (Last 12 months)0
  • Downloads (Last 6 weeks)0
Reflects downloads up to 25 Nov 2024

Other Metrics

Citations

Cited By

View all
  • (2024)Unveiling Delay Effects in Traffic Forecasting: A Perspective from Spatial-Temporal Delay Differential EquationsProceedings of the ACM Web Conference 202410.1145/3589334.3645688(1035-1044)Online publication date: 13-May-2024
  • (2024)On the maximum principle and high-order, delay-free integrators for the viscous Cahn–Hilliard equationAdvances in Computational Mathematics10.1007/s10444-024-10143-650:3Online publication date: 3-May-2024

View Options

View options

Login options

Media

Figures

Other

Tables

Share

Share

Share this Publication link

Share on social media