A new scaling for Newton's iteration for the polar decomposition and its backward stability

SIAM Journal on Matrix Analysis and Applications

Volumn 30, Issue 2, 2008, Pages 822-843

  Byers, Ralph ^a   Xu, Hongguo ^a  

^a UNIVERSITY OF KANSAS   (United States)

Author keywords

Matrix sign function; Newton's method; Numerical stability; Polar decomposition; Scaling; Singular value decomposition (SVD)

Indexed keywords

MATRIX SIGN FUNCTION; NEWTON'S METHOD; NUMERICAL STABILITY; POLAR DECOMPOSITION; SCALING; SINGULAR VALUE DECOMPOSITION (SVD);

CONVERGENCE OF NUMERICAL METHODS; NEWTON-RAPHSON METHOD; NUMBER THEORY;

SINGULAR VALUE DECOMPOSITION;

EID: 67649206981     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/070699895     Document Type: Article

References (23)

1. Z. BAI and J. DEMMEL, Design of a parallel nonsymmetric eigenroutine toolbox, Part I. in Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientific Computing, R. F. Sincovec et al., eds., SIAM, Philadelphia, 1993, pp. 391-398. Also available as Computer Science Report CSD-92-718, University of California, Berkeley, CA 1992.
2. Z. BAI and J. DEMMEL, Design of a Parallel Nonsymmetric Eigenroutine Toolbox, Part II, Technical report Department of Mathematics Research Report 95-11, University of Kentucky, Lexington, KY, 1995.
3. L. A. BALZER, Accelerated convergence of the matrix sign function method of solving Lyapunov, Riccati and other matrix equations, Internat. J. Control, 32 (1980), pp. 1057-1078.
4. R. BYERS, Solving the algebraic Riccati equation with the matrix sign function, Linear Algebra Appl., 85 (1987), pp. 267-279.
5. A. A. DUBRULLE, An optimum iteration for the matrix polar decomposition, Electron. Trans. Numer. Anal., 8 (1999), pp. 21-25 (electronic).
6. W. GANDER, Algorithms for the polar decomposition, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 1102-1115.
7. G. H. GOLUB and C. F. VAN LOAN, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
8. N. J. HIGHAM, Computing the polar decomposition-with applications, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 1160-1174.
9. N. J. HIGHAM, Computing a nearest symmetric positive semidefinite matrix, Linear Algebra Appl., 103 (1988), pp. 103-118.
10. N. J. HIGHAM, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, 2002.
11. N. J. HIGHAM, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, 2008.
12. N. J. HIGHAM and P. PAPADIMITRIOU, Parallel Singular Value Decomposition via the Polar Decomposition, Technical report Numerical Analysis Report 239, Department of Mathematics, University of Manchester, Manchester, England, 1993.
13. N. J. HIGHAM and P. PAPADIMITRIOU, A parallel algorithm for computing the polar decomposition, Parallel Comput., 20 (1994), pp. 1161-1173.
14. N. J. HIGHAM and R. S. SCHREIBER, Fast polar decomposition of an arbitrary matrix, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 648-655.
15. R. A. HORN and C. R. JOHNSON, Matrix Analysis, Cambridge University Press, Cambridge, 1990, corrected reprint of the 1985 original.
16. C. S. KENNEY and A. J. LAUB, On scaling Newton's method for polar decomposition and the matrix sign function, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 688-706.
17. C. S. KENNEY and A. J. LAUB, The matrix sign function, IEEE Trans. Automat. Control, 40 (1995), pp. 1330-1348.
18. A. KIELBASIŃSKI and K. ZIȨTAK, Numerical behavior of Higham's scaled method for polar decomposition, Numer. Algorithms, 32 (2003), pp. 105-140.
19. A. KIELBASIŃSKI, P. ZIELIŃSKI, and K. ZIȨTAK, Higham's scaled method for polar decomposition and numerical matrix-inversion, Technical report Institute of Mathematics and Computer Science Report I18/2007/P-045, Wroclaw University of Technology, Wroclaw, Poland, 2007.
20. R.-C. LI, New perturbation bounds for the unitary polar factor, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 327-332.
21. R. MATHIAS, Perturbation bounds for the polar decomposition, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 588-597.
22. J. D. ROBERTS, Linear model reduction and solution of the algebraic Riccati equation by use of the sign function, Internat. J. Control, 32 (1980), pp. 677-687. (Reprint of Technical report TR-13, CUED/B-Control, Cambridge University, Engineering Department, Cambridge, 1971).
23. P. ZIELIŃSKI and K. ZIȨTAK, Polar decomposition-properties, applications and algorithms, Appl. Math., An. Polish Math. Soc., 38 (1995), pp. 23-49.