Accurate computation of the product-induced singular value decomposition with applications

SIAM Journal on Numerical Analysis

Volumn 35, Issue 5, 1998, Pages 1969-1994

  Drmač, Zlatko ^a  

^a UCB 450   (United States)

Author keywords

Contragredient transformation; Eigenvalue problem; Product induced singular value decomposition; Relative accuracy; Singular value decomposition; System balancing

Indexed keywords

EID: 0000075074     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142995292633     Document Type: Article

References (39)

1. E. ANDERSON, Z. BAI, C. BISCHOF, J. DEMMEL, J. DONGARRA, J. D. CROZ, A. GREENBAUM, S. HAMMARLING, A. MCKENNY, S. OSTROUCHOV, AND D. SORENSEN, LAPACK Users' Guide, 2nd ed., SIAM, Philadelphia, 1992.
2. J. BARLOW, Stability analysis of the G-algorithm and a note on its application to sparse least squares problems, BIT, 25 (1985), pp. 507-520.
3. A. BJÖRCK, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996.
4. P. A. BUSINGER AND G. H. GOLUB, Linear least squares solutions by Householder transformations, Numer. Math., 7 (1965), pp. 269-276.
5. P. P. M. DE RIJK, A one-sided Jacobi algorithm for computing the singular value decomposition on a vector computer, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 359-371.
6. J. DEMMEL, S. EISENSTAT, M. GU, I. SLAPNIČAR, AND K. VESELIĆ, Notes on Computing the SVD with High Relative Accuracy, preprint, 1995.
7. J. DEMMEL AND A. MCKENNEY, A Test Matrix Generation Suite, LAPACK Working Note 9, Courant Institute, New York, 1989.
8. J. DEMMEL AND K. VESELIĆ, Jacobi's method is more accurate than QR, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 1204-1245.
9. Z. DRMAČ, Computing the Singular and the Generalized Singular Values, Ph.D. thesis, Lehrgebiet Mathematische Physik, Fernuniversität Hagen, Hagen, Germany, 1994.
10. Z. DRMAČ, A tangent algorithm for computing the generalized singular value decomposition, SIAM J. Numer. Anal., 35 (1998), pp. 1804-1832.
11. Z. DRMAČ, On the condition behaviour in the Jacobi method, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 509-514.
12. Z. DRMAČ, Implementation of Jacobi rotations for accurate singular value computation in floating point arithmetic, SIAM J. Sci. Comput., 18 (1997), pp. 1200-1222.
13. Z. DRMAČ, M. OMLADIČ, AND K. VESELIĆ, On the perturbation of the Cholesky factorization, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 1319-1332.
14. Z. DRMAČ AND K. VESELIĆ, Note on the Accuracy of the Eigenvalues or Singular Values of Matrices Generated by Finite Elements, preprint, 1995.
15. S. EISENSTAT AND I. IPSEN, Relative perturbation techniques for singular value problems, SIAM J. Numer. Anal., 32 (1995), pp. 1972-1988.
16. K. V. FERNANDO AND S. HAMMARLING, A product induced singular value decomposition (πSVD) for two matrices and balanced realization, in Linear Algebra in Signals, Systems, and Control, SIAM, Philadelphia, 1988, pp. 128-140.
17. W. M. GENTLEMAN, Error analysis of QR decompositions by Givens transformations, Linear Algebra Appl., 10 (1975), pp. 189-197.
18. S. K. GODUNOV, A. G. ANTONOV, O. P. KIRILYUK, AND V. I. KOSTIN, Garantirovannaya Tochnost Resheniya Sistem Lineǐnykh Uravneniǐ v Evklidovykh Prostranstvakh, Nauka, Sibirskoe Otdelenie, Novosibirsk, 1988.
19. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations 2nd ed., The Johns Hopkins University Press, Baltimore, MD, 1989.
20. M. GU AND S. EISENSTAT, Efficient algorithms for computing a strong rank-revealing QR factorization, SIAM J. Sci. Comput., 17 (1996), pp. 848-869.
21. S. HAMMARLING, Numerical solution of the stable, non-negative definite Lyapunov equation, IMA J. Numer. Anal., 2 (1982), pp. 303-323.
22. M. T. HEATH, A. J. LAUB, C. C. PAIGE, AND R. C. WARD, Computing the singular value decomposition of a product of two matrices, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 1147-1159.
23. M. R. HESTENES, Inversion of matrices by biorthogonalization and related results, J. SIAM, 6 (1958), pp. 51-90.
24. N. J. HIGHAM, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, 1996.
25. R. A. HORN AND C. R. JOHNSON, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.
26. C. G. J. JACOBI, Über ein leichtes Verfahren die in der Theorie der Säcularstörungen vorkommenden Gleichungen numerisch aufzulösen, Crelle's Journal für reine und angew. Math., 30 (1846), pp. 51-95.
27. A. J. LAUB, M. T. HEATH, C. C. PAIGE, AND R. C. WARD, Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms, IEEE Trans. Automat. Control, AC-32 (1987), pp. 115-122.
28. R.-C. LI, Relative Perturbation Theory: (I) Eigenvalue and Singular Value Variations, Technical Report, Mathematical Science Section, Oak Ridge National Laboratory, Oak Ridge, TN, 1996; SIAM J. Matrix Anal. Appl., 19 (1998), pp. 956-982.
29. R.-C. LI, Relative Perturbation Theory: (I) Eigenvalue and Singular Value Variations, Technical Report, Mathematical Science Section, Oak Ridge National Laboratory, Oak Ridge, TN, 1996; SIAM J. Matrix Anal. Appl., 19 (1998), pp. 956-982.
30. W. F. MASCARENHAS, A note on Jacobi being more accurate than QR, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 215-218.
31. R. MATHIAS, Spectral Perturbation Bounds for Positive Definite Matrices, Technical Report, Department of Mathematics, College of William and Mary, Williamsburg, VA, 1995.
32. J. J. MODI AND M. R. B. CLARKE, An alternative Givens ordering, Numer. Math., 43 (1984), pp. 83-90.
33. J. R. RICE, A theory of condition, SIAM J. Numer. Anal., 3 (1966), pp. 287-310.
34. H. RUTISHAUSER, Vorlesungen über numerische Mathematik, Band 2, Differentialgleichungen und Eigenwertprobleme, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften, Math. Reihe, Band 57, Birkhäuser Verlag, Basel, Stuttgart, 1976.
35. A. VAN DER SLUIS, Condition numbers and equilibration of matrices, Numer. Math., 14 (1969), pp. 14-23.
36. K. VESELIĆ AND V. HARI, A note on a one-sided Jacobi algorithm, Numer. Math., 56 (1989), pp. 627-633.
37. K. VESELIĆ AND I. SLAPNIČAR, Floating-point perturbations of Hermitian matrices, Linear Algebra Appl., 195 (1993), pp. 81-116.
38. J. H. WILKINSON, Error analysis of direct methods of matrix inversion, J. Assoc. Comput. Mach., 8 (1962), pp. 281-330.
39. J. H. WILKINSON, The Algebraic Eigenvalue Problem, Springer-Verlag, Berlin, Heidelberg, New York, 1965.