A Schur algorithm for computing matrix pth roots

SIAM Journal on Matrix Analysis and Applications

Volumn 24, Issue 4, 2003, Pages 971-989

  Smith, Matthew I ^a  

^a UNIVERSITY OF MANCHESTER   (United Kingdom)

Author keywords

Commutativity; Matrix pth root; Newton's method; Real arithmetic; Rounding error analysis; Schur algorithm; Stability

Indexed keywords

EID: 0142188203     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479801392697     Document Type: Article

References (23)

1. A. BJÖRCK AND S. HAMMARLING, A Schur method for the square root of a matrix, Linear Algebra Appl., 52/53 (1983), pp. 127-140.
2. Ç. K. KOÇ AND B. BAKKALOǦLU, Halley's method for the matrix sector function, IEEE Trans. Automat. Control, 40 (1995), pp. 944-949.
3. S. H. CHENG, N. J. HIGHAM, C. S. KENNEY, AND A. J. LAUB, Approximating the logarithm of a matrix to specified accuracy, SIAM J. Matrix Anal. Appl., 22 (2001), pp. 1112-1125.
4. J. H. CURRY, L. GARNETT, AND D. SULLIVAN, On the iteration of a rational function: Computer experiments with Newton's method, Comm. Math. Phys., 91 (1983), pp. 267-277.
5. J. W. DEMMEL, Applied Numerical Linear Algebra, SIAM, Philadelphia, PA, 1997.
6. J. E. DENNIS, JR., AND R. B. SCHNABEL, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, NJ, 1983.
7. P. FATOU, Sur les équations fonctionnelles, Bull. Soc. Math. France, 47 (1919), pp. 161-271.
8. F. R. GANTMACHER, The Theory of Matrices, Vol. I, Chelsea, New York, 1959.
9. G. H. GOLUB AND C. F. V. LOAN, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
10. N. J. HIGHAM, Newton's method for the matrix square root, Math. Comp., 46 (1986), pp. 537549.
11. N. J. HIGHAM, Computing real square roots of a real matrix, Linear Algebra Appl., 88/89 (1987), pp. 405-430.
12. N. J. HIGHAM, The Test Matrix Toolbox for MATLAB (Version 3.0), Numerical Analysis Report 276, Manchester Centre for Computational Mathematics, Manchester, England, 1995.
13. N. J. HIGHAM, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, PA, 1996.
14. W. D. HOSKINS AND D. J. WALTON, A faster, more stable method for computing the pth roots of positive definite matrices, Linear Algebra Appl., 26 (1979), pp. 139-163.
15. G. JULIA, Mémoire sur l'itération des fonctions rationelles, J. Math. Pure Appl., 8 (1918), pp. 47-245.
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. 2 - I = 0, Math. Tables Aids Comput., 12 (1958), pp. 109-116.
18. J. M. ORTEGA, Numerical Analysis: A Second Course, Academic Press, New York, 1972.
19. L. S. SHIEH, Y. T. TSAY, AND C. T. WANG, Matrix sector functions and their applications to systems theory, Proc. IEE-D, 131 (1984), pp. 171-181.
20. G. W. STEWART, Introduction to Matrix Computations, Academic Press, London, 1973.
21. F. TISSEUR, Newton's method in floating point arithmetic and iterative refinement of generalized eigenvalue problems, SIAM J. Matrix Anal. Appl., 22 (2001), pp. 1038-1057.
22. E. R. VRSCAY, Julia sets and Mandlebrot-like sets associated with higher order Schröder rational iteration functions: A computer assisted study, Math. Comp., 46 (1996), pp. 151-169.
23. J. H. M. WEDDERBURN, Lectures on Matrices, AMS, Providence, RI, 1934.