A Schur-Fréchet algorithm for computing the logarithm and exponential of a matrix

SIAM Journal on Matrix Analysis and Applications

Volumn 19, Issue 3, 1998, Pages 640-663

  Kenney, C S ^a   Laub, A J ^b  

^a University of California   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Matrix exponential; Matrix functions; Matrix logarithm

Indexed keywords

EID: 0032371908     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479896300334     Document Type: Article

References (31)

1. G. A. BAKER, Essentials of Padé Approximants, Academic Press, New York, 1975.
2. A. BJÖRCK AND S. HAMMARLING, A Schur method for the square root of a matrix, Linear Algebra Appl., 52/53 (1983), pp. 127-140.
3. F. BRAUER AND J. A. NOHEL, The Qualitative Theory of Ordinary Differential Equations, W.A. Benjamin, New York, 1969.
4. C. H. EDWARDS, JR., The Historical Development of the Calculus, Springer-Verlag, New York, 1979.
5. W. CULVER, On the existence and uniqueness of the real logarithm of a matrix, in Proc. Amer. Math. Soc., 17 (1966), pp. 1146-1151.
6. L. DIECI, B. MORINI, AND A. PAPINI, Computational techniques for real logarithms of matrices, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 570-593.
7. M. EISENBUD, Environmental Radioactivity, McGraw-Hill, New York, 1963.
8. F. R. GANTMACHER, The Theory of Matrices, Vol. I, Chelsea, New York, 1959.
9. H. H. GOLDSTINE, A History of Numerical Analysis from the 16th through the 19th Century, Springer-Verlag, New York, 1977.
10. I. GRADSHTEYN AND I. RYZHIK, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965.
11. A. GRAHAM, Kronecker Products and Matrix Calculus with Applications, John Wiley, New York, 1981.
12. B. HELTON, Logarithms of matrices, Proc. Amer. Math. Soc., 19 (1968), pp. 733-738.
13. N. J. HIGHAM, Computing real square roots of a real matrix, Linear Algebra Appl., 88/89 (1987), pp. 405-430.
14. R. A. HORN AND C. R. JOHNSON, Topics in Matrix Analysis, Cambridge University Press, New York, 1991.
15. B. KÅGSTRÖM, Numerical Computation of Matrix Functions, Tech. report UMINF-58.77, Department of Information Processing, University of Umeå, Umeå, Sweden, 1977.
16. C. S. KENNEY AND A. J. LAUB, Condition estimates for matrix functions, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 191-209.
17. C. S. KENNEY AND A. J. LAUB, Padé error estimates for the logarithm of a matrix, Internat. J. Control, 50 (1989), pp. 707-730.
18. C. S. KENNEY AND A. J. LAUB, Small-sample statistical condition estimates for general matrix functions, SIAM J. Sci. Comp., 15 (1994), pp. 36-61.
19. C. S. KENNEY AND A. J. LAUB, The matrix sign function, IEEE Trans. Automat. Control, 40 (1995), pp. 1330-1348.
20. C. S. KENNEY AND A. J. LAUB, A Schur-Fréchet Algorithm for the Logarithm of a Matrix, Tech. report, Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA, 1995.
21. G. J. LASTMAN AND N. K. SINHA, Infinite series for logarithm of matrix applied to identification of linear continuous-time multivariable systems from discrete-time models, Electron. Lett., 27 (1991), pp. 1468-1470.
22. Y. L. LUKE, The Special Functions and Their Approximations, Vols. I and II, Academic Press, New York, 1969.
23. R. MATHIAS, Evaluating the Fréchet derivative of the matrix exponential, Numer. Math., 63 (1992), pp. 213-226.
24. R. MATHIAS, Condition estimation for matrix functions via the Schur decomposition, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 565-578.
25. R. MATHIAS, Approximation of matrix-valued functions, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 1061-1063.
26. C. B. MOLER AND C. F. VAN LOAN, Nineteen dubious ways to compute the exponential of a matrix, SIAM Rev., 20 (1978), pp. 801-836.
27. I. NAJFELD AND T. F. HAVEL, Derivatives of the matrix exponential and their computation, Adv. Appl. Math., 16 (1995), pp. 321-375.
28. B. N. PARLETT, A recurrence among the elements of functions of triangular matrices, Linear Algebra Appl., 14 (1976), pp. 117-121.
29. N. K. SINHA AND G. J. LASTMAN, Transformation algorithm for identification of continuous-time multivariable systems from discrete data, Electron. Lett., 17 (1981), pp. 779-780.
30. R. C. WARD, Numerical computation of the matrix exponential with accuracy estimate, SIAM J. Numer. Anal., 14 (1977), pp. 600-614.
31. A. WOUK, Integral representation of the logarithm of matrices and operators, J. Math. Anal. Appl., 11 (1965), pp. 131-138.