Approximating the logarithm of a matrix to specified accuracy

SIAM Journal on Matrix Analysis and Applications

Volumn 22, Issue 4, 2001, Pages 1112-1125

  Cheng, Sheung Hun ^a   Higham, Nicholas J ^a   Kenney, Charles S ^b   Laub, Alan J ^c  

^a UNIVERSITY OF MANCHESTER   (United Kingdom)

^b University of California   (United States)

^c University of California   (United States)

Author keywords

Denman Beavers iteration; Inverse scaling and squaring method; Matrix logarithm; Matrix square root; Pad approximation

Indexed keywords

EID: 0035537402     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479899364015     Document Type: Article

References (21)

1. R. C. ALLEN AND S. A. PRUESS, An analysis of an inverse problem in ordinary differential equations, SIAM J. Sci. Statist. Comput., 2 (1981), pp. 176-185.
2. G. A. BAKER, JR., Essentials of Padé Approximants, Academic Press, New York, 1975.
3. G. A. BAKER, JR. AND P. GRAVES-MORRIS, Padé Approximants, Encyclopedia Math. Appl., 2nd ed., Cambridge University Press, Cambridge, England, 1996.
4. A. BJÖRCK AND S. HAMMARLING, A Schur method for the square root of a matrix, Linear Algebra Appl., 52/53 (1983), pp. 127-140.
5. E. D. DENMAN AND A. N. BEAVERS, JR., The matrix sign function and computations in systems, Appl. Math. Comput., 2 (1976), pp. 63-94.
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. L. DIECI AND A. PAPINI, Conditioning and Padé approximation of the logarithm of a matrix, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 913-930.
8. N. J. HIGHAM, Newton's method for the matrix square root, Math. Comp., 46 (1986), pp. 537-549.
9. N. J. HIGHAM, Computing real square roots of a real matrix, Linear Algebra Appl., 88/89 (1987), pp. 405-430.
10. N. J. HIGHAM, The matrix sign decomposition and its relation to the polar decomposition, Linear Algebra Appl., 212/213 (1994), pp. 3-20.
11. N. J. HIGHAM, Stable iterations for the matrix square root, Numer. Algorithms, 15 (1997), pp. 227-242.
12. N. J. HIGHAM, A New sqrtm for MATLAB, Numerical Analysis Report 336, Manchester Centre for Computational Mathematics, Manchester, England, January 1999.
13. N. J. HIGHAM, Evaluating Padé approximants of the matrix logarithm, SIAM J. Matrix Anal. Appl., 22 (2001), pp. 1126-1135.
14. R. A. HORN AND C. R. JOHNSON, Topics in Matrix Analysis, Cambridge University Press, Cambridge, England, 1991.
15. C. KENNEY AND A. J. LAUB, Condition estimates for matrix functions, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 191-209.
16. C. KENNEY AND A. J. LAUB, Padé error estimates for the logarithm of a matrix, Internat. J. Control, 50 (1989), pp. 707-730.
17. C. KENNEY AND A. J. LAUB, Rational iterative methods for the matrix sign function, SIAM J. Matrix Anal. Appl., 12 (1991), pp. 273-291.
18. C. S. KENNEY AND A. J. LAUB, The matrix sign function, IEEE Trans. Automat. Control, 40 (1995), pp. 1330-1348.
19. 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.
20. B. SINGER AND S. SPILERMAN, The representation of social processes by Markov models, Amer. J. Sociology, 82 (1976), pp. 1-54.
21. R. C. WARD, Numerical computation of the matrix exponential with accuracy estimate, SIAM J. Numer. Anal., 14 (1977), pp. 600-610.