Approximation of matrices and a family of gander methods for polar decomposition

BIT Numerical Mathematics

Volumn 46, Issue 2, 2006, Pages 345-366

  Laszkiewicz, B ^a   Zietak K ^a  

^a WROCLAW UNIVERSITY OF TECHNOLOGY   (Poland)

Author keywords

Approximation by subunitary matrices; Family of Gander methods; Higham's scaled method; Minimal rank approximation; Polar decomposition

Indexed keywords

EID: 33745300149     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10543-006-0053-4     Document Type: Article

References (26)

1. A. Ben-Israel and D. Cohen, On iterative computation of generalized inverses and associated projections, SIAM J. Numer. Anal., 3 (1966), pp. 410-419.
2. A. Ben-Israel and T. N. E. Greville, Generalized Inverses. Theory and Applications, Springer, New York, 2003.
3. R. Bhatia, Some inequalities for norm ideals, Commun. Math. Phys., 111 (1987), pp. 33-39.
4. R. Bhatia, Matrix Analysis, Springer, Berlin, 1996.
5. Å. Björck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996.
6. C. Eckart and G. Young, The approximation of one matrix by another of lower rank, Psychometrika 1 (1936), pp. 211-218.
7. K. Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA, 37 (1951), pp. 760-766.
8. K. Fan and A. Hoffman, Some metric inequalities in the space of matrices, Proc. Am. Math. Soc., 6 (1955), pp. 111-116.
9. W. Gander, Algorithms for the polar decomposition, SIAM J. Sci. Stat. Comput., 11 (1990), pp. 1102-1115.
10. G. H. Golub, Least squares, singular values and matrix approximations, Aplikace Matematiky, 13 (1968), pp. 44-51.
11. P. R. Halmos, Spectral approximation of normal operators, Proc. Edinb. Math. Soc., II. Ser., 19 (1974), pp. 51-58.
12. N. J. Higham, Computing the polar decomposition - with applications, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 1160-1174.
13. N. J. Higham, Matrix nearness problem and applications, in Applications of Matrix Theory, M. J. C. Gover and S. Barnett, eds., pp. 1-27, Oxford University Press, Oxford, 1990.
14. N. J. Higham, D. S. Mackey, N. Mackey, and F. Tisseur, Computing the polar decomposition and the matrix sign decomposition in matrix groups, SIAM J. Matrix Anal. Appl., 25 (2004), pp. 1178-1192.
15. N. J. Higham and R. S. Schreiber, Fast polar decomposition of an arbitrary matrix, SIAM J. Sci. Stat. Comput., 11 (1990), pp. 648-655.
16. R. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, 1991.
17. C. Kenney and A. J. Laub, On scaling Newton method for polar decomposition and the matrix sign function, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 688-706.
18. A. Kielbasiński and K. Ziȩtak, Numerical behaviour of Higham's scaled method for polar decomposition, Numer. Algorithms, 32 (2003), pp. 105-140.
19. D. S. Mackey, N. Mackey, and F. Tisseur, Structured factorizations in scalar product spaces, SIAM J. Matrix Anal. Appl., 27 (2006), pp. 821-850.
20. P. J. Maher, Partially isometric approximation of positive operators, Illinois J. Math., 33 (1989), pp. 227-243.
21. P. J. Maher, Some matrix approximation problems arising from quantum chemistry, Proc. Indian Natl. Sci. Acad., 64 (1998), pp. 715-723.
22. C. R. Rao, Matrix approximation and reduction of dimensionality in multivariate statistical analysis, in Multivariate Analysis V, Proc. of the Fifth Int. Symposium on Multivariate Analysis, pp. 3-22, North-Holland, Amsterdam, 1980.
23. G. W. Stewart, Matrix Algorithms, Vol. I: Basic Decompositions, SIAM, Philadelphia, 1998.
24. J.-G. Sun and C.-H. Chen, Generalized polar decomposition, Math. Numer. Sinica, 11 (1989), pp. 262-273 (in Chinese).
25. P. Y. Wu, Approximation by partial isometries, Proc. Edinb. Math. Soc., 29 (1986), pp. 255-261.
26. P. Zieliński and K. Ziȩtak, The polar decomposition - properties, applications and algorithms, Applied Mathematics, Ann. Pol. Math. Soc., 38 (1995), pp. 23-49.