Matrix displacement decompositions and applications to Toeplitz linear systems

Linear Algebra and Its Applications

Volumn 268, Issue 1-3, 1998, Pages 197-225

  Di Fiore, Carmine ^a   Zellini, Paolo ^b  

^a SAPIENZA UNIVERSITY OF ROME   (Italy)

^b UNIVERSITY OF ROME TOR VERGATA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0041355748     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0024-3795(97)00044-X     Document Type: Article

References (39)

1. R. C. Agarwal and C. S. Burrus, Fast one-dimensional digital convolution by multidimensional techniques, IEEE Trans. Acoust. Speech Signal Process. 22:1-10 (1974).
2. R. C. Agarwal and J. W. Cooley, New algorithms for digital convolution, IEEE Trans. Acoust. Speech Signal Process. 25:392-410 (1977).
3. L. J. Alex, Diophantine equations related to finite groups, Comm. Algebra 4:77-100 (1976).
4. G. Ammar and P. D. Gader, New decompositions of the inverse of a Toeplitz matrix, in Signal Processing, Scattering and Operator Theory, and Numerical Methods, Proceedings of the International Symposium MTNS-89 (M. A. Kaashoek, J. H. van Schuppen, and A. C. N. Ran, Eds.), Birkhäuser, Boston, 1990, Vol. III, pp. 421-428.
5. G. Ammar and P. D. Gader, A variant of the Gohberg-Semencul formula involving circulant matrices, SIAM J. Matrix Anal. Appl. 12:534-540 (1991).
6. R. B. Bapat and V. S. Sunder, On hypergroups of matrices, Linear and Multilinear Algebra 29:125-140 (1991).
7. D. Bini, On a class of matrices related to Toeplitz matrices, Technical Report 83-5, Computer Science Dept., SUNYA, Albany, N.Y., 1983.
8. D. Bini, Parallel solution of certain Toeplitz systems, SIAM J. Comput. 13:268-276 (1984).
9. D. Bini and M. Capovani, Spectral and computational properties of band symmetric Toeplitz matrices, Linear Algebra Appl. 52/53:99-126 (1983).
10. D. Bini and M. Capovani, Fast parallel and sequential computations and spectral properties concerning band Toeplitz matrices, Calcolo 20:177-189 (1983).
11. D. Bini and M. Capovani, Tensor rank and border rank of band Toeplitz matrices, SIAM J. Comput. 16:252-258 (1987).
12. D. Bini and P. Favati, On a matrix algebra related to discrete Hartley transform, SIAM J. Matrix Anal. Appl. 14:500-507 (1993).
13. D. Bini and V. Pan, Improved parallel computations with Toeplitz-like and Hankel-like matrices, Linear Algebra Appl. 188/189:3-29 (1993).
14. D. Bini and V. Pan, Polynomial and Matrix Computations, Vol. 1 Fundamental Algorithms, Birkhäuser, Boston, 1994.
15. R. E. Blahut, Fast Algorithms for Digital Signal Processing, Addison-Wesley, Reading, Mass., 1987.
16. E. Bozzo, Algebras of higher dimension for displacement decompositions and computations with Toeplitz plus Hankel matrices, Linear Algebra Appl. 230:127-150 (1995).
17. E. Bozzo, Matrix Algebras and Discrete Transforms, Ph.D. Thesis, TD-1/94, Univ. di Pisa, Pisa, 1994.
18. E. Bozzo and C. Di Fiore, On the use of certain matrix algebras associated with discrete trigonometric transforms in matrix displacement decomposition, SIAM J. Matrix Anal. Appl. 16:312-326 (1995).
19. P. J. Davis, Circulant Matrices, Wiley, New York, 1979.
20. B. M. M. De Weger, Solving exponential diophantine equations using lattice basis reduction algorithms, J. Number Theory 26:325-367 (1987).
21. C. Di Fiore and P. Zellini, Matrix decompositions using displacement rank and classes of commutative matrix algebras, Linear Algebra Appl. 229:49-99 (1995).
22. B. Friedlander, M. Morf, T. Kailath, and L. Ljung, New inversion formulas for matrices classified in terms of their distance from Toeplitz matrices, Linear Algebra Appl. 27:31-60 (1979).
23. P. D. Gader, Displacement operator based decompositions of matrices using circulants or other group matrices, Linear Algebra Appl. 139:111-131 (1990).
24. I. Gohberg and V. Olshevsky, Circulants, displacements and decompositions of matrices, Integral Equations Operator Theory 15:730-743 (1992).
25. I. Gohberg and A. Semencul, On the inversion of finite Toeplitz matrices and their continual analogues (in Russian), Mat. Issled. 7(2):201-233 (1972).
26. G. Heinig and K. Rost, Algebraic Methods for Toeplitz-like Matrices and Operators, Akademie-Verlag, Berlin, and Birkhäuser, Boston, 1984.
27. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge U.P., New York, 1993.
28. J. R. Jain, An efficient algorithm for a large Toeplitz set of linear equations, IEEE Trans. Acoust. Speech Signal Process. 27:612-615 (1979).
29. T. Kailath and J. Chun, Generalized displacement structure for block-Toeplitz, Toeplitz-block, and Toeplitz-derived matrices, SIAM J. Matrix Anal. Appl. 15:114-128 (1994).
30. T. Kailath, S. Kung, and M. Morf Displacement ranks of matrices and linear equations, J. Math. Anal. Appl. 68:395-407 (1979).
31. V. Pan, Complexity of computations with matrices and polynomials, SIAM Rev. 34:225-262 (1992).
32. W. Press, B. Flannery, S. Tenkolsky, and W. Vetterling, Numerical Recipes - the Art of Scientific Computing, Cambridge U.P., 1992.
33. A. H. Sayed, H. Lev-ari, and T. Kailath, Fast triangular factorization of the sum of quasi-Toeplitz and quasi-Hankel matrices, Linear Algebra Appl. 191:77-106 (1993).
34. T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge U.P., London, 1986.
35. P. N. Swarztrauber, Symmetric FFTs, Math. Comp. 47:323-346 (1986).
36. Z. Wang, Fast algorithms for the discrete W transform and for the discrete Fourier transform, IEEE Trans. Acoust. Speech Signal Process. 32:803-816 (1984).
37. S. Winograd, On computing the discrete Fourier transform, Math. Comp. 32:175-199 (1978).
38. S. Winograd, Arithmetic Complexity of Computations, CBMS-NSF Regional Conf. Ser. in Appl. Math., SIAM, Philadelphia, 1980.
39. P. Zellini, On the optimal computation of a set of symmetric and persymmetric bilinear forms, Linear Algebra Appl. 23:101-119 (1979).