A new scaling and squaring algorithm for the matrix exponential

SIAM Journal on Matrix Analysis and Applications

Volumn 31, Issue 3, 2009, Pages 970-989

  Al Mohy, Awad H ^a   Higham, Nicholas J ^a  

^a UNIVERSITY OF MANCHESTER   (United Kingdom)

Author keywords

Backward error analysis; Expm; MATLAB; Matrix exponential; Matrix function; Matrix norm estimation; Overscaling; Pad approximation; Scaling and squaring method

Indexed keywords

BACKWARD ERROR ANALYSIS; MATRIX EXPONENTIAL; MATRIX EXPONENTIALS; MATRIX NORMS; SCALING AND SQUARING METHOD;

ALGORITHMS; DIGITAL ARITHMETIC; ERROR ANALYSIS; ESTIMATION; FUNCTION EVALUATION; LEAST SQUARES APPROXIMATIONS; MATLAB;

MATRIX ALGEBRA;

EID: 71549127525     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/09074721X     Document Type: Article

References (20)

1. A. H. AL-MOHY AND N. J. HIGHAM, Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation, SIAM J. Matrix Anal. Appl., 30 (2009), pp. 1639-1657.
2. M. BOSSA, E. ZACUR, AND S. OLMOS, Algorithms for computing the group exponential of diffeomorphisms: Performance evaluation, in Proceedings of the Computer Vision and Pattern Recognition Workshops, 2008 (CVPRW '08), IEEE Computer Society, Washington, DC, 2008, pp. 1-8.
3. P. I. DAVIES AND N. J. HIGHAM, A Schur-Parlett algorithm for computing matrix functions, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 464-485.
4. L. DIECI AND A. PAPINI, Padé approximation for the exponential of a block triangular matrix, Linear Algebra Appl., 308 (2000), pp. 183-202.
5. L. DIECI AND A. PAPINI, Conditioning of the exponential of a block triangular matrix, Numer. Algorithms, 28 (2001), pp. 137-150.
6. G. I. HARGREAVES AND N. J. HIGHAM, Efficient algorithms for the matrix cosine and sine, Numer. Algorithms, 40 (2005), pp. 383-400.
7. N. J. HIGHAM, The Matrix Computation Toolbox, available online at http://www.ma.man.ac.uk/~higham/mctoolbox.
8. N. J. HIGHAM, The Matrix Function Toolbox, available online at http://www.ma.man.ac.uk/~higham/mftoolbox.
9. N. J. HIGHAM, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, 2002.
10. N. J. HIGHAM, The scaling and squaring method for the matrix exponential revisited, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 1179-1193.
11. N. J. HIGHAM, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, 2008.
12. N. J. HIGHAM AND F. TISSEUR, A block algorithm for matrix 1-norm estimation, with an application to 1-norm pseudospectra, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1185-1201.
13. R. A. HORN AND C. R. JOHNSON, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.
14. C. S. KENNEY AND A. J. LAUB, A Schur-Fréchet algorithm for computing the logarithm and exponential of a matrix, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 640-663.
15. L. MORAI AND A. F. PACHECO, Algebraic approach to the radioactive decay equations, Amer. J. Phys., 71 (2003), pp. 684-686.
16. R. B. SIDJE, Expokit: A software package for computing matrix exponentials, ACM Trans. Math. Software, 24 (1998), pp. 130-156.
17. The Control and Systems Library SLICOT, available online at http://www.slicot.org/.
18. C. F. VAN LOAN, Computing integrals involving the matrix exponential, IEEE Trans. Automat. Control, 23 (1978), pp. 395-404.
19. T. G. WRIGHT, Eigtool, 2002. Available online at http://www.comlab.ox.ac. uk/pseudospectra/eigtool/.
20. D. YUAN AND W. KERNAN, Explicit solutions for exit-only radioactive decay chains, J. Appl. Phys., 101 (2007), 094907.