On Logarithmic Norms 论文
摘要
Previous article Next article On Logarithmic NormsTorsten StrömTorsten Strömhttps://doi.org/10.1137/0712055PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractThis 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.[1] F. L. Bauer, , J. Stoer and , C. Witzgall, Absolute and monotonic norms, Numer. Math., 3 (1961), 257–264 10.1007/BF01386026 MR0130104 0111.01602 CrossrefGoogle Scholar[2] R. Bellman, Stability Theory of Differential Equations, McGraw-Hill, New York, 1965 Google Scholar[3] Richard Bellman, Introduction to matrix analysis, McGraw-Hill Book Co., Inc., New York, 1960xx+328 MR0122820 0124.01001 Google Scholar[4] G. J. Cooper, Error bounds for numerical solutions of ordinary differential equations, Numer. Math., 18 (1971/72), 162–170 10.1007/BF01436325 MR0301938 0252.65059 CrossrefISIGoogle Scholar[5] W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Co., Boston, Mass., 1965viii+166 MR0190463 0154.09301 Google Scholar[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 Google Scholar[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Wis., 1966), John Wiley & Sons Inc., New York, 1966, 89–96 MR0208838 0161.12403 Google Scholar[8] G. Dahlquist, 1966, Unpublished lecture notes, Atlas Computer Labs., Didcot, England Google Scholar[9] Charles A. Desoer and , Hiromasa Haneda, The measure of a matrix as a tool to analyze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 MR0453292 CrossrefISIGoogle Scholar[10] Bo Einarsson, Numerical treatment of integro-differential equations with a certain maximum property, Numer. Math., 18 (1971/72), 267–288 10.1007/BF01397086 MR0307513 0233.73168 CrossrefISIGoogle Scholar[11] L. Elsner, Verfahren zur Berechnung des Spektralradius nichtnegativer irreduzibler Matrizen, Computing (Arch. Elektron. Rechnen), 8 (1971), 32–39 MR0293829 0225.65051 Google Scholar[12] John H. George and , Robert W. Gunderson, Conditioning of linear boundary value problems, Nordisk Tidskr. Informationsbehandling (BIT), 12 (1972), 172–181 MR0309317 0243.65045 Google Scholar[13] K. G. Guderley and , C. L. Keller, A basic theorem in the computation of ellipsoidal error bounds, Numer. Math., 19 (1972), 218–229 10.1007/BF01404692 MR0321308 0227.65046 CrossrefISIGoogle Scholar[14] Peter Henrici, Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices, Numer. Math., 4 (1962), 24–40 10.1007/BF01386294 MR0135706 0102.01502 CrossrefGoogle Scholar[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 MR0176616 0196.49903 Google Scholar[16] Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964xi+257 MR0175290 0161.12101 Google Scholar[17] W. Kahan, An ellipsoidal error bound for linear systems of ordinary differential equations, SIAM Rev., 8 (1966), 568–569 ISIGoogle Scholar[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.) Google Scholar[19] S. M. Lozinskij, A contribution to the theory of f nice matrices, Soviet Math. Dokl., 163 (1965), 1014–1016 0192.13202 Google Scholar[20] S. M. Lozinskij, Estimates of a spherical matrix norm and of the corresponding logarithmic norm, Soviet Math. Dokl., 165 (1965), 1511–1514 0161.02904 Google Scholar[21] E. E. Osborne, On pre-conditioning of matrices, J. Assoc. Comput. Mach., 7 (1960), 338–345 MR0143333 0106.31604 CrossrefISIGoogle Scholar[22] C. V. Pao, Logarithmic derivates of a square matrix, Linear Algebra and Appl., 6 (1973), 159–164 10.1016/0024-3795(73)90015-3 MR0320037 0246.15021 CrossrefGoogle Scholar[23] C. V. Pao, A further remark on the logarithmic derivatives of a square matrix, Linear Algebra and Appl., 7 (1973), 275–278 10.1016/0024-3795(73)90044-X MR0325654 0257.15016 CrossrefGoogle Scholar[24] B. Parlett and , 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. 10.1007/BF02165404 0184.37703 CrossrefISIGoogle Scholar[25] Torsten Ström, On logarithmic norms, Department of Information Processing and Computer Science, The Royal Institute of Technology, Stockholm, 1969i+8 MR0343574 Google Scholar[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 Google Scholar[27] T. Ström, Minimization of norms and logarithmic norms by diagonal similarities, Computing (Arch. Elektron. Rechnen), 10 (1972), 1–7 MR0314253 0251.65034 Google Scholar[28] J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965xviii+662, London MR0184422 0258.65037 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo modelApplied Numerical Mathematics, Vol. 179 | 1 Sep 2022 Cross Ref Generalization of the Multiplicative and Additive Compounds of Square Matrices and Contraction Theory in the Hausdorff DimensionIEEE Transactions on Automatic Control, Vol. 67, No. 9 | 1 Sep 2022 Cross Ref A Sufficient Condition for $k$-Contraction of the Series Connection of Two SystemsIEEE Transactions on Automatic Control, Vol. 67, No. 9 | 1 Sep 2022 Cross Ref Optimal control of variable-speed wind turbines modeled as Markov jump systemsJournal of the Franklin Institute, Vol. 359, No. 10 | 1 Jul 2022 Cross Ref On the prescribed-time attractivity and frozen-time eigenvalues of linear time-varying systemsAutomatica, Vol. 140 | 1 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