Representations of Toeplitz-plus-Hankel matrices using trigonometric transformations with application to fast matrix-vector multiplication

Linear Algebra and Its Applications

Volumn 275-276, Issue , 1998, Pages 225-248

  Heinig, Georg ^a   Rost, Karla ^b  

^a KUWAIT UNIVERSITY   (Kuwait)

^b CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Cosine transform; Fast algorithm; Hankel matrix; Toeplitz matrix

Indexed keywords

EID: 0032388205     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0024-3795(97)10024-6     Document Type: Article

References (28)

1. D. Bini, P. Favati, On a matrix algebra related to the discrete Hartley transform, SIAM J. Matrix Anal. Appl. 14 (1993) 500-507.
2. D. Bini, V. Pan, Polynomial and Matrix Computations, I. Fundamental Algorithms, Birkhäuser, Boston, MA, 1994.
3. E. Boman, I. Koltracht, Fast transform based preconditioner for Toeplitz equations, SIAM J. Matrix Anal. Appl. 16 (1995) 628-645.
4. E. Bozzo, Algebras of higher dimension for displacement decompositions and computations with Toeplitz-plus-Hankel matrices, Linear Algebra Appl. 230 (1995) 127-150.
5. E. Bozzo, C. di Fiore, On the use of certain matrix algebras associated with discrete trigonometric transforms in matrix displacement decompositions, SIAM J. Matrix Anal. Appl. 16 (1995) 312-326.
6. R. Chan, K. Ng, Conjugate gradient method for Toeplitz systems, SIAM Review 38 (1996) 427-482.
7. P.J. Davis, Circulant Matrices, Wiley, New York, 1979.
8. P. Duhamel, M. Vetterli. Improved Fourier and Hartley transform algorithms, IEEE Trans. Acoust. Speech Signal Process. 35 (1987) 849-863.
9. C. di Fiore, P. Zellini, Matrix decompositions using displacement rank and classes of commutative matrix algebras, Linear Algebra Appl. 229 (1995) 49-99.
10. I. Gohberg. V. Olshevsky, Fast algorithms with preprocessing for matrix-vector multiplication, J. Complexity 10 (1994) 411-427.
11. I. Gohberg, V. Olshevsky, Complexity of multiplication with vectors for structured matrices, Linear Algebra Appl. 202 (1994) 163-192.
12. G. Heinig, Transformation techniques for fast and stable solution of Toeplitz systems and polynomial equations, in: Proceedings of the International Workshop on Recent Advances in Applied Mathematics, Kuwait University, May 1996, pp. 223-238.
13. G. Heinig, A. Bojanczyk, Transformation approaches for structured matrices, I. Transformations, Linear Algebra Appl. 254 (1997) 193-226.
14. T. Huckle, Cauchy matrices and iterative methods for Toeplitz matrices, in: Proceedings of SPIE-95, vol. 2563. 1995, pp. 281-292.
15. T. Kailath, V. Olshevsky, Displacement structure approach to discrete-trigonometric-transform based preconditioners of G. Strang type and of T. Chan type, Linear Algebra Appl.
16. Hao Lu, Generalized Trummer's problem with application to Vandermonde-like systems, SIAM J. Matrix Anal. Appl. 254 (1997) 193-226.
17. V. Olshevski, A fast real-arithmetic algorithm for multiplication of a triangular Toeplitz matrix by a vector, Preprint 1993 (unpublished).
18. W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes. The Art of Scientific Computing, Cambridge Univ. Press, Cambridge, 1996.
19. L.D. Pustilnikov, The algebraic structure of spaces of Toeplitz and Hankel matrices (Russian), Dokl. Akad. Nauk SSSR 250 (3) (1980) 556-559.
20. K.R. Rao, P. Yip, Discrete Cosine Transform: Algorithms, Advantages, and Applications, Academic Press, Boston, MA, 1990.
21. H. Sorensen, D. Jones, M. Heidman, C. Burrus, Real-valued fast Fourier transform algorithms, Signal Processing 6 (1984) 267-278.
22. G. Steidl, Fast radix-p discrete cosine transform, Appl. Algebra Eng. Commun. Comput. 3 (1992) 39-46.
23. G. Steidl, M. Tasche, A polynomial approach to fast algorithms for discrete Fourier-cosine and Fourier-sine transforms, Math. Comp. 56 (1991) 281-296.
24. G. Steidl, M. Tasche, Private communication.
25. M. Tasche, Fast algorithms for discrete Chebyshev-Vandermonde transforms and applications, Numerical Algorithms 5 (1993) 453-464.
26. M. Tismenetsky, A decomposition of Toeplitz matrices and optimal circulant pre-conditioning, Linear Algebra Appl. 154-156 (1991) 105-121.
27. C. van Loan, Computational Framework for the Fast Fourier Transform, SIAM, Philadelphia, PA, 1992.
28. M. Vetterli, H. Nussbaumer, Simple FFT and DCT algorithms with reduced number of operations, Signal Process. 6 (1984) 267-278.