Recursive-based PCG methods for Toeplitz systems with nonnegative generating functions

SIAM Journal on Scientific Computing

Volumn 24, Issue 5, 2003, Pages 1507-1529

  Ng, Michael K ^a   Sun, Hai Wei ^b   Jin, Xiao Qing ^c  

^a UNIVERSITY OF HONG KONG   (Hong Kong)

^b GUANGDONG UNIVERSITY OF TECHNOLOGY   (China)

^c UNIVERSITY OF MACAU   (Macao)

Author keywords

Gohberg Semencul formula; Preconditioned conjugate gradient method; Preconditioners; Recursive based method; Toeplitz matrices

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; FUNCTIONS; MATRIX ALGEBRA; POLES AND ZEROS; PROBLEM SOLVING;

GOHBERG-SEMENCUL FORMULA; LEVINSON-DURBIN METHOD; PRECONDITIONED CONJUGATE GRADIENT METHOD; RECURSIVE-BASED METHOD; TOEPLITZ MATRICES;

GRADIENT METHODS;

EID: 0141940605     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827500378155     Document Type: Article

References (43)

1. G. S. Ammar and W. B Gragg, Superfast solution of real positive definite Toeplitz systems, SIAM J. Matrix Anal. Appl., 9 (1988), pp. 61-76.
2. G. Ammar and P. Gader, New decomposition of the inverse of a Toeplitz matrix, in Signal Processing, Scattering and Operator Theory, and Numerical Methods, Proceedings of the International Symposium on the Mathematical Theory of Networks and Systems Vol. 3, M. Kaashoek, J. van Schuppen, and A. Ran, eds., Birkhäuser, Boston, 1990, pp. 421-428.
3. F. Di Benedetto, Analysis of preconditioning techniques for ill-conditioned Toeplitz matrices, SIAM J. Sci. Statist. Comput., 16 (1995), pp. 682-697.
4. F. Di Benedetto and S. Serra, A unifying approach to abstract matrix algebra preconditioning, Numer. Math., 82 (1999), pp. 57-90.
5. R. Bitmead and B. Anderson, Asymptotically fast solution of Toeplitz and related systems of linear equations, Linear Algebra Appl., 34 (1980), pp. 103-116.
6. R. Brent, F. Gustavson, and D. Yun, Fast solution of Toeplitz systems of equations and computation of Padé approximants, J. Algorithms, 1 (1980), pp. 259-295.
7. R. Chan, Toeplitz preconditioners for Toeplitz systems with nonnegative generating functions, IMA J. Numer. Anal., 11 (1991), pp. 333-345.
8. R. H. Chan, Q.-S. Chang, and H.-W. Sun, Multigrid method for ill-conditioned symmetric Toeplitz systems, SIAM J. Sci. Comput., 19 (1998), pp. 516-529.
9. R. H. Chan and K.-P. Ng, Fast iterative solvers for Toeplitz-plus-band systems, SIAM J. Sci. Statist. Comput., 14 (1993), pp. 1013-1019.
10. R. H. Chan and M. K. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38 (1996), pp. 427-482.
11. R. Chan, M. Ng, and A. Yip, The best circulant preconditioners for Hermitian Toeplitz systems II: The multiple-zero case, Numer. Math., to appear.
12. R. H. Chan and M.-C. Yeung, Circulant preconditioners constructed from kernels, SIAM J. Numer. Anal., 29 (1992), pp. 1093-1103.
13. R. H. Chan, A. M. Yip, and M. K. Ng, The best circulant preconditioners for Hermitian Toeplitz systems, SIAM J. Numer. Anal., 38 (2001), pp. 876-896.
14. T. F. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 766-771.
15. J. Durbin, The fitting of time series models, Rev. Int. Stat., 28 (1960), pp. 233-244.
16. G. Fiorentino and S. Serra, Multigrid methods for symmetric positive definite block Toeplitz matrices with nonnegative generating functions, SIAM J. Sci. Comput., 17 (1996), pp. 1068-1081.
17. I. Gohberg and A. Semencul, On the inversion of finite Toeplitz matrices and their continuous analogs, Mat. Issled., 2 (1972), pp. 201-233.
18. I. Gohberg and G. Heinig, Inversion of finite-section Toeplitz matrices consisting of elements of non-commutative algebra, Rev. Roumaine Math. Pures Appl., 19 (1974), pp. 623-663.
19. G. Heinig and L. Rost, Algebraic Methods for Toeplitz-like Matrices and Operators, Birkhäuser Verlag, Basel, Switzerland, 1984.
20. G. Heinig and L. Rost, Hartley transform representations of symmetric Toeplitz matrix inverses with application to fast matrix-vector multiplications, SIAM J. Matrix Anal. Appl., 22 (2000), pp. 86-105.
21. F. de Hoog, A new algorithm for solving Toeplitz systems of equations, Linear Algebra Appl., 88/89 (1987), pp. 123-138.
22. S. Haykin, Adaptive Filter Theory, Prentice-Hall, London, 1991.
23. T. Huckle, Iterative Methods for Toeplitz-like Matrices, Report SCCM-94-05, Computer Science Department, Stanford University, Stanford, CA, 1994.
24. T. Huckle, Superfast solution of linear equations with low displacement rank, in High Performance Algorithms for Structured Matrix Problems, Adv. Theory Comput. Comput. Math. II, P. Arbenz, M. Paprzycki, A. H. Sameh, and V. Sarin, eds., Nova Sci. Publ., Commack, NY, 1998, pp. 149-162.
25. T. Huckle, Multigrid preconditioning and Toeplitz matrices, Electron. Trans. Numer. Anal., 13, (2002), pp. 81-105.
26. R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.
27. N. Levinson, The Wiener RMS (root mean square) error criterion in filter design and prediction, J. Math. Phys., 25 (1947), pp. 261-278.
28. M. Morf, Doubling algorithms for Toeplitz and related equations, in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Process., 1980, pp. 954-959.
29. J. Olkin, Linear and Nonlinear Deconvolution Problems, Ph.D. thesis, Rice University, Houston, TX, 1986.
30. D. Potts and G. Steidl, Preconditioners for ill-conditioned Toeplitz systems constructed from positive kernels, SIAM J. Sci. Comput., 22 (2002), pp. 1741-1761.
31. S. Serra, Preconditioning strategies for Hermitian Toeplitz systems with nondefinite generating functions, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 1007-1019.
32. S. Serra, On the extreme eigenvalues of Hermitian (block) Toeplitz matrices, Linear Algebra Appl., 270 (1998), pp. 109-129.
33. S. Serra, Korovkin theorems and linear positive gram matrix algebra approximations of Toeplitz matrices, Linear Algebra Appl., 284 (1998), pp. 307-334.
34. S. Serra, Spectral and computational analysis of block Toeplitz matrices having nonnegative definite matrix-valued generating functions, BIT, 39, (1999), PP. 152-175.
35. S. Serra, Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs matrix-sequences, Numer. Math., 92 (2002), pp. 433-465.
36. S. Serra Capizzano and E. Tyrtyshnikov, Any circulant-like preconditioner for multilevel matrices is not superlinear, SIAM J. Matrix Anal. Appl., 21 (1999), pp. 431-439.
37. S. Serra, How to choose the best iterative strategy for symmetric Toeplitz systems, SIAM J. Numer. Anal., 36 (1999), pp. 1078-1103.
38. G. Strang, A proposal for Toeplitz matrix calculations, Stud. Appl. Math., 74 (1986), pp. 171-176.
39. T. Strohmer, A Levinson-Galerkin algorithm for regularized trigonometric approximation, SIAM J. Sci. Comput., 22 (2000), pp. 1160-1183.
40. H. Sun, X. Jin, and Q. Chang, Convergence of the multigrid method for ill-conditioned block Toeplitz systems, BIT, 41 (2001), pp. 179-190.
41. W. F. Trench, An algorithm for the inversion of finite Toeplitz matrices, SIAM J. Appl. Math., 12 (1964), pp. 515-522.
42. E. Tyrtyshnikov, Circulant preconditioners with unbounded inverses, Linear Algebra Appl., 216 (1995), pp. 1-24.
43. M. Yeung and R. Chan, Circulant preconditioners for Toeplitz matrices with piecewise continuous generating functions, Math. Comp., 61 (1993), pp. 701-718.