Iterative methods for X - AXB = C

Journal of Computational and Applied Mathematics

Volumn 86, Issue 1, 1997, Pages 73-101

  Calvetti, D ^a   Levenberg, N ^b   Reichel, L ^c  

^a CASE WESTERN RESERVE UNIVERSITY   (United States)

^b UNIVERSITY OF AUCKLAND   (New Zealand)

^c KENT STATE UNIVERSITY   (United States)

Author keywords

Alternating direction; Iterative methods; Matrix equations

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; IMAGE RECONSTRUCTION; MATRIX ALGEBRA;

ALTERNATING DIRECTION IMPLICIT ITERATIVE METHODS;

ITERATIVE METHODS;

EID: 0031270563     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(97)00149-0     Document Type: Article

References (28)

1. G.S. Ammar, W.B. Gragg, The generalized Schur algorithm for the superfast solution of Toeplitz systems, in Rational Approximation and its Applications in Mathematics and Physics, J. Gilewicz, M. Pindor, W. Siemaszko (Eds.), Lecture Notes in Math. vol. 1237, Springer, Berlin, 1987, pp. 315-330.
2. G.S. Ammar, W.B. Gragg, Numerical experience with a superfast real Toeplitz solver, Linear Algebra Appl. 121 (1989) 185-206.
3. H.C. Andrews, B.R. Hunt, Digital Image Restoration, Prentice-Hall, Englewood Cliffs, NJ, 1977.
4. T. Bagby, The modulus of a plane condenser, J. Math. Mech. 17 (1967) 315-329.
5. T. Bagby, On interpolation by rational functions, Duke J. Math. 36 (1969) 95-104.
6. R. Bartels, G.W. Stewart, Algorithm 432: Solution of the matrix equation AX + XB = C, Comm. ACM 15 (1972) 820-826.
7. D. Calvetti, L. Reichel, Application of ADI iterative methods to the restoration of noisy images, SIAM J. Matrix Anal. Appl. 17 (1996) 165-186.
8. P.L.C. Cheong, S.D. Morgera, Iterative methods for restoring noisy images, IEEE Trans. Acoust. Speech Signal Proc. 37 (1989) 580-585.
9. B.N. Datta, K. Datta, Theoretical and computational aspects of some linear algebra problems in control theory, in: C.I. Byrnes, A. Lindquist (Eds.), Computational and Combinatorial Methods in Systems Theory, Elsevier, Amsterdam, 1986, pp. 201-212.
10. N.S. Ellner, E.L. Wachspress, Alternating direction implicit iteration for systems with complex spectra, SIAM J. Numer. Anal. 28 (1991) 859-870.
11. F.R. Gantmacher, Matrizentheorie, Springer, New York, 1986.
12. G.H. Golub, S. Nash, C. Van Loan, A Hessenberg-Schur method for the problem AX + XB = C, IEEE Trans. Automat. Control. AC-24 (1979) 909-913.
13. G.H. Golub, C.F. Van Loan, Matrix Computations, 2nd ed., Johns Hopkins University Press, Baltimore, 1989.
14. R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
15. D.Y. Hu, L. Reichel, Krylov subspace methods for the Sylvester equation, Linear Algebra Appl. 172 (1992) 283-313.
16. M. Kac, W.L. Murdock, G. Szego, On the eigenvalues of certain Hermitian forms, J. Rat. Mech. Anal. 2 (1953) 767-800.
17. R.L. Lagendijk, J. Biemond, Iterative Identification and Restoration of Images, Kluwer, Dordrecht, 1991.
18. N.S. Landkof, Foundations of Potential Theory, Springer, Berlin, 1972.
19. N. Levenberg, L. Reichel, A generalized ADI iterative method, Numer. Math. 66 (1993) 215-233.
20. A.L. Levin, E.B. Saff, Optimal ray sequences of rational functions connected with the Zolotarev problem, Constr. Approx. 10 (1994) 235-273. Erratum, Constr. Approx. 12 (1996) 437-438.
21. Y. Saad, Numerical solution of large Lyapunov equations, in: M.A. Kaashoek, J.H. van Schuppen, A.C.M. Ran (Eds.), Signal Processing, Scattering, Operator Theory and Numerical Methods, Birkhäuser, Boston, MA, 1990, pp. 503-511.
22. H. Stahl, V. Totik, General Orthogonal Polynomials, Cambridge University Press, Cambridge, 1992.
23. G. Starke, Optimal alternating direction implicit parameters for nonsymmetric systems of linear equations, SIAM J. Numer. Anal. 28 (1991) 1431-1445.
24. G. Starke, Fejér-Walsh points for rational functions and their use in the ADI iterative method, J. Comput. Appl. Math. 46 (1993) 129-141.
25. M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.
26. R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.
27. E.L. Wachspress, Iterative solution of the Lyapunov matrix equation, Appl. Math. Lett. 1 (1988) 87-90.
28. E.L. Wachspress, The ADI Model Problem, Windsor, 1995.